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\begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash pagestyle{fancy} \end_layout \begin_layout Plain Layout \backslash fancyhf{} \end_layout \begin_layout Plain Layout \backslash fancyhead[RE,RO]{ \backslash scshape \backslash nouppercase{ \backslash textit{ \backslash leftmark}}} \end_layout \begin_layout Plain Layout % \backslash fancyhead[LE,LO]{ \backslash textit{Nonlinear dynamics \backslash & stochastic processes in cybersecurity applications}} \end_layout \begin_layout Plain Layout \backslash fancyfoot[LE,LO]{ \backslash textit{Pierce Ryan}} \end_layout \begin_layout Plain Layout \backslash fancyfoot[RE,RO]{ \backslash thepage} \end_layout \begin_layout Plain Layout \backslash renewcommand{ \backslash footrulewidth}{0.4pt}% Line at the footer visible \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash fancypagestyle{plain}{% \end_layout \begin_layout Plain Layout \backslash fancyhf{}% \end_layout 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\backslash \backslash \end_layout \begin_layout Plain Layout \backslash vspace{1.5cm} \end_layout \begin_layout Plain Layout \backslash includegraphics[scale=0.2]{ucc_crest.jpg} \backslash \backslash \end_layout \begin_layout Plain Layout \backslash vspace{.25cm} \end_layout \begin_layout Plain Layout { \backslash Large \backslash textbf{National University of Ireland, Cork} \backslash \backslash \end_layout \begin_layout Plain Layout \backslash textit{ \backslash Large Ollscoil na h \backslash 'Eireann, Corcaigh}} \backslash \backslash \end_layout \begin_layout Plain Layout \backslash vspace{1cm} \end_layout \begin_layout Plain Layout { \backslash large \backslash textsc{Department of Applied Mathematics}} \backslash \backslash \end_layout \begin_layout Plain Layout \backslash vspace{1cm} \end_layout \begin_layout Plain Layout { \backslash Large Thesis submitted for the degree of \backslash \backslash \end_layout \begin_layout Plain Layout \backslash LARGE \backslash textbf{Doctor of Philosophy}} \backslash \backslash \end_layout \begin_layout Plain Layout \backslash vspace{1.5cm} \end_layout \begin_layout Plain Layout { \backslash large Awarded $19^{ \backslash text{th}}$ June 2023} \backslash \backslash \end_layout \begin_layout Plain Layout \backslash vspace{1.5cm} \end_layout \begin_layout Plain Layout \backslash begin{large} \end_layout \begin_layout Plain Layout \backslash begin{tabular}{rl} \end_layout \begin_layout Plain Layout Head of Department:& \backslash quad Prof. Sebastian Wieczorek \backslash \backslash \end_layout \begin_layout Plain Layout & \backslash \backslash \end_layout \begin_layout Plain Layout Supervisors:& \backslash quad Dr. Andreas Amann \backslash \backslash \end_layout \begin_layout Plain Layout & \backslash quad Dr. Sorcha Healy \end_layout \begin_layout Plain Layout \backslash end{tabular} \end_layout \begin_layout Plain Layout \backslash end{large} \end_layout \begin_layout Plain Layout \backslash end{center} \end_layout \begin_layout Plain Layout \backslash end{titlepage} \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash frontmatter \end_layout \begin_layout Plain Layout \backslash addtocontents{toc}{ \backslash protect \backslash thispagestyle{empty}} \end_layout \begin_layout Plain Layout \backslash tableofcontents \end_layout \begin_layout Plain Layout \backslash thispagestyle{plain} \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash newpage \end_layout \begin_layout Plain Layout \backslash vspace*{ \backslash fill} \end_layout \begin_layout Plain Layout \backslash renewcommand{ \backslash leftmark}{Declaration} \end_layout \begin_layout Plain Layout \backslash noindent \end_layout \begin_layout Plain Layout This is to certify that the work I am submitting is my own and has not been submitted for another degree, either at University College Cork or elsewhere. All external references and sources are clearly acknowledged and identified within the contents. I have read and understood the regulations of University College Cork concernin g plagiarism. \backslash \backslash \end_layout \begin_layout Plain Layout \backslash vspace{3cm} \backslash \backslash \end_layout \begin_layout Plain Layout \backslash begin{tabular}{@{}p{.5in}p{5in}@{}} \end_layout \begin_layout Plain Layout & \backslash hrulefill \backslash \backslash \end_layout \begin_layout Plain Layout & \backslash \backslash \end_layout \begin_layout Plain Layout & Pierce Ryan \backslash \backslash \end_layout \begin_layout Plain Layout \backslash end{tabular} \end_layout \begin_layout Plain Layout \backslash vspace*{ \backslash fill} \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout % \backslash newpage \end_layout \begin_layout Plain Layout % \backslash vspace*{6cm} \end_layout \begin_layout Plain Layout % \backslash begin{center} \end_layout \begin_layout Plain Layout % \backslash textit{``There are two types of thesis: The ones that are perfect, and the ones that are done.''} \end_layout \begin_layout Plain Layout % \backslash end{center} \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout % \backslash vspace*{ \backslash fill} \end_layout \begin_layout Plain Layout \backslash newpage \end_layout \begin_layout Plain Layout \backslash renewcommand{ \backslash leftmark}{Dedication} \end_layout \begin_layout Plain Layout \backslash vspace*{6cm} \end_layout \begin_layout Plain Layout \backslash begin{center} \end_layout \begin_layout Plain Layout \backslash textit{For my grandfathers.} \end_layout \begin_layout Plain Layout \backslash end{center} \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash onehalfspacing \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout \backslash cleardoublepage \end_layout \begin_layout Plain Layout % \backslash phantomsection \end_layout \begin_layout Plain Layout % \backslash addcontentsline{toc}{chapter}{ \backslash listfigurename} \end_layout \begin_layout Plain Layout % \backslash listoffigures \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout % \backslash cleardoublepage \end_layout \begin_layout Plain Layout % \backslash phantomsection \end_layout \begin_layout Plain Layout % \backslash addcontentsline{toc}{chapter}{ \backslash listtablename} \end_layout \begin_layout Plain Layout % \backslash listoftables \end_layout \begin_layout Plain Layout \end_layout \begin_layout Plain Layout % \backslash cleardoublepage \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \begin_layout Chapter* Acknowledgements \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash phantomsection \end_layout \begin_layout Plain Layout \backslash addcontentsline{toc}{chapter}{Acknowledgements} \end_layout \begin_layout Plain Layout \backslash renewcommand{ \backslash leftmark}{Acknowledgements} \end_layout \end_inset \end_layout \begin_layout Standard There are many people without whom I would not have finished this thesis, and many more without whom I would not have finished this thesis with my sanity largely still intact. \end_layout \begin_layout Standard To start with, I thank my supervisors and mentors, Andreas, Sorcha and Niall, for their guidance. Five years ago, two old friends from the Tyndall Institute randomly ran into each other outside a sports centre. Sorcha had moved towards industry, becoming a data scientist at the cybersecuri ty provider McAfee. Andreas had moved towards academia, becoming a lecturer at University College Cork. Sorcha brought up a hybrid industry/academia funded postgraduate programme she wanted to pursue, and asked whether Andreas had any ideas for projects. A few months later, I was writing a proposal for the Irish Research Council Employment-Based Postgraduate Programme with Sorcha and Andreas. At McAfee, Sorcha and I worked side by side for over a year before Covid cleared us out of the office, while my weekly meetings with Andreas at UCC would range from an hour during the winter to \begin_inset Quotes bld \end_inset until we run out of things to talk about' during the summer. When Sorcha's career took her from McAfee to Microsoft in the final year of the project, Niall stepped in to be my mentor at McAfee. Without their patience, guidance and humour, I wouldn't have gotten this far. I'd also like to thank my monitor, Kathleen, for her help and the use of her extensive library of statistics textbooks. \end_layout \begin_layout Standard I thank my family for their support; my parents Eoin and Sheelagh, my sister Aisling and her husband Aidan. Throughout the COVID-19 pandemic, while we bubbled together in Baltimore, they ignored the sound of my head banging off my desk with tact and discretion. Sheelagh, as she coordinated local HSE efforts against the pandemic, inspired me with her unwavering patience. Eoin provided endless supplies of wisdom and outstanding food, while Aidan could always be relied on for a laugh. Aisling, who like me, followed Sheelagh in pursuing a PhD, even shared an office with me, which we christened the UCC Southwest-est Campus. \end_layout \begin_layout Standard I thank my colleagues at McAfee for their acceptance. When I first began working at McAfee at the start of this project, there were plenty of people curious as to what my job actually was. I was one of them. Still, they welcomed me with open arms. Sorcha, Gerard, Niamh, Jill, Niall and I made up the Applied Data Science team. Together we weathered five changes in management, and countless sprint planning meetings. For a while, I practically had Gerard on speed dial to answer my \begin_inset Quotes bld \end_inset data science for dummies' questions. Other standouts include Paul, Didier, Rachel, Jon, Crystal, and last but certainly not least, Brian, my final and longest serving/suffering manager. \end_layout \begin_layout Standard Finally, I'd like to thank my friends. My fellow PhD students Andrew, Chris, Eoin, Rory and Conor, who brought that windowless office on the second floor to life, and my friends Ciara, Matt and \begin_inset ERT status open \begin_layout Plain Layout \backslash 'E \end_layout \end_inset adaoin, who made sure I went to the pub on a regular basis no matter how much or how little I protested. \end_layout \begin_layout Standard This research was supported by the Irish Research Council and McAfee LLC through the Irish Research Council Employment-Based Postgraduate Programme (grant number EBPPG/2018/269). \end_layout \begin_layout Chapter* Abstract \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash phantomsection \end_layout \begin_layout Plain Layout \backslash addcontentsline{toc}{chapter}{Abstract} \end_layout \begin_layout Plain Layout \backslash renewcommand{ \backslash leftmark}{Abstract} \end_layout \end_inset The Internet is an extremely complex system which has a significant impact on the world we live in. In this thesis, we formalise Internet-based problems as mathematical models to better understand their dynamics. Modelling these problems requires dynamical features such as time delay, periodic forcing, switching and stochasticity. We study several dynamical systems which employ a combination of these features from Internet applications, including targeted ransomware, data networks, and signal processing. We also study a climate science system which shares features with the signal processing system and exhibits similar dynamics. Stochasticity is found to be critical in the modelling of the negotiations involved in targeted ransomware, while time delay is a crucial feature in the modelling of data networks. The signal processing and climate science systems give rise to extremely rich dynamics, which we are able to study analytically due to the presence of switching. This yields further insights into related smooth systems. \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash mainmatter \end_layout \end_inset \end_layout \begin_layout Chapter* Introduction \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout % \backslash phantomsection \end_layout \begin_layout Plain Layout \backslash addcontentsline{toc}{chapter}{Introduction} \end_layout \begin_layout Plain Layout \backslash renewcommand{ \backslash leftmark}{Introduction} \end_layout \end_inset \end_layout \begin_layout Standard The Internet is the most complex man-made system in the world, with its dynamics driven by the activities of both man and machine. The basic infrastructure, which was established in the 1970s with ARPANET \begin_inset CommandInset citation LatexCommand cite key "gillies2000web" literal "false" \end_inset , has grown in capacity, complexity and utility through the iterations of Web 1.0, Web 2.0, and Web 3.0 \begin_inset CommandInset citation LatexCommand cite key "shivalingaiah2008comparative,fuchs2010theoretical" literal "false" \end_inset . The Internet has expanded to support the broadest repository of information, the largest platform for international trade, and the greatest conduit of human interaction. But it is much more than just a system. The Internet is practically a world unto itself, a mirror of our own physical world which functions according to its own logic. \end_layout \begin_layout Standard In the physical world, your social network is constrained by physical distance and language barriers. On the Internet, physical distance translates to a fraction of a second of latency and language barriers are broken down by instant translation, enabling your social circle to span the globe. At the same time, the Internet is subject to effects which do not exist in the real world. As you walk down the street, shops don't reshuffle to move those that paid higher advertising fees right in front of you. Your peripheral vision is not filled by advertisements for products you talked about yesterday. There are no \begin_inset Quotes bld \end_inset bots' pasting fake reviews on the windows of businesses, and no algorithms subtly guiding you to places where you'll spend the most money. \end_layout \begin_layout Standard The Internet is a world separate to our own, and yet they are irrevocably intertwined. The Internet is affected by events in the physical world. The basic infrastructure runs on physical hardware, chunks of doped silicon connected by wire and fibre optic, any piece of which may fail at any moment, taking fragments of the Internet with it. Many actions taken in the physical world, malicious or benign, start with, or are facilitated by, communication over the Internet. Online threats such as \emph on ransomware \emph default cause real-world harm through the leaking of private data, the extortion of ransoms, and the proliferation of criminal organisations financed by these crimes \begin_inset CommandInset citation LatexCommand citep key "beek2016targeted,coveware2020ransomware,kalaimannan2017influences" literal "false" \end_inset . Even people who do not engage with the Internet are subject to its influence, and so understanding the dynamics of the digital world is crucial to understand ing our own. \end_layout \begin_layout Standard Due to the scale and complexity of the Internet, there is a truly vast amount to be understood. The Internet is a patchwork of systems and subsystems which fulfil a list of functions too long to fit in any thesis. As a result of this diverse array of functions, the Internet exhibits a broad range of dynamics across multiple scales. \end_layout \begin_layout Standard At the large scale, we observe network phenomena and the collective dynamics of millions of devices connected by the Internet. When we focus on the connections between personal profiles on social media websites such as Facebook, we observe social networks. These networks have been found to exhibit small-world and scale-free properties \begin_inset CommandInset citation LatexCommand cite key "watts1998collective,albert2002statistical" literal "false" \end_inset . \end_layout \begin_layout Standard Less visible than social media websites are the background systems which silently propagate information between servers and personal computers. These \emph on data networks \emph default make use of various techniques to increase efficiency and minimise latency to the point that people forget these systems are even there. One of the most ubiquitous of these techniques is \emph on caching \emph default , where data sent to a computer is stored temporarily in case it is needed again in quick succession. Dynamically speaking, caching has two effects. The first is that it induces hysteresis, explicit dependence of a system on its history. The second is that the cost of using data from other computers changes discontinuously depending on whether or not the data is cached. Both hysteresis and discontinuous response give rise to rich dynamical behaviour, such as multistability \begin_inset CommandInset citation LatexCommand citep key "larger_virtual_2013,shayer2000stability,foss1996multistability,kim1997multistability,keane_delayed_2015" literal "true" \end_inset and border-collision bifurcations \begin_inset CommandInset citation LatexCommand cite key "barton2005explicit,banerjee_border_1999,di_bernardo_bifurcations_2008,colombo_bifurcations_2012,barton2006periodic" literal "false" \end_inset , respectively. \end_layout \begin_layout Standard At the small scale, we can consider the activity of a single computer interactin g with a data network. Here, the effects of the physical world become far more apparent. From an Internet perspective, the activity of humans can appear random; while some of their actions can be predicted based on their previous activity on the Internet, some actions are prompted by events in the real world. From the perspective of the Internet, this introduces stochasticity into their dynamical behaviour. In addition, while machines don't sleep, most people do, usually according to a regular sleep cycle. Therefore we can expect that any dynamical process which requires human interaction to feature both periodicity and stochasticity. Again, both periodicity \begin_inset CommandInset citation LatexCommand citep key "boccaletti_synchronization_2018,marchionne_synchronisation_2018,wieczorek_dynamical_2005,tziperman_nino_1994" literal "true" \end_inset and stochasticity \begin_inset CommandInset citation LatexCommand citep key "selten1988simple,pikovsky1997coherence,barndorff2001non" literal "true" \end_inset are known to induce dynamical behaviour, including synchronization and resonance. \end_layout \begin_layout Standard The core principle of this thesis is to formalise small-scale problems on the Internet in mathematical terms to gain a deeper understanding of these problems and establish a foundation for further research at all scales. Specifically, we will study the developments in ransomware which have led to the current wave of targeted ransomware, and consider how dynamical features such as time delay, periodic forcing, stochasticity and switching can appear in data networks. We will also study dynamical systems from climate science and signal processing which share dynamical features with those seen in Internet systems. \end_layout \begin_layout Section* Thesis structure \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash phantomsection \end_layout \begin_layout Plain Layout \backslash addcontentsline{toc}{section}{Thesis structure} \end_layout \end_inset \end_layout \begin_layout Standard \series bold Chapter \begin_inset CommandInset ref LatexCommand ref reference "chap:Border-collision-bifurcations-in" plural "false" caps "false" noprefix "false" \end_inset \series default is largely reproduced from P. Ryan, A. Keane and A. Amann, \emph on Border-collision bifurcations in a driven time-delay system \emph default , Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(2):023121, 2020. \begin_inset CommandInset citation LatexCommand cite key "ryan2020border" literal "false" \end_inset . We study a piecewise-smooth time-delay system with periodic forcing derived from a phenomenological climate model \begin_inset CommandInset citation LatexCommand citep key "ghil_delay_2008" literal "true" \end_inset which demonstrates an extremely complex resonance structure. In particular, our system is piecewise-constant, as both the time-delayed feedback and the periodic forcing are switches, flipping back and forth between two values. This yields an extremely simple system which still demonstrates much of the same dynamics as the original model. We develop a symbolic representation for the non-smooth dynamics of the system, and explore the bifurcations of the system through bifurcations of Poincaré maps derived from the symbolic dynamics. These include both traditional bifurcations and border-collision bifurcations. This symbolic representation enables us to analytically calculate solutions and bifurcation curves while studying phenomena that were previously studied only numerically. This work has been found to be of use in more recent research as the study of non-smooth systems continues to develop \begin_inset CommandInset citation LatexCommand cite key "belykh2023beyond,agliari2022appearance" literal "false" \end_inset . \begin_inset ERT status open \begin_layout Plain Layout \backslash newline \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash noindent \end_layout \end_inset \series bold Chapter \begin_inset CommandInset ref LatexCommand ref reference "chap:Dynamics-of-a" plural "false" caps "false" noprefix "false" \end_inset \series default forms the basis of a paper that I am co-authoring with Dr. Lucas Illing and Dr. Andreas Amann. We study a piecewise-linear second-order delay differential equation which is representative of feedback systems with relays that actuate after a fixed time delay. This system demonstrates a complex structure of classical and border-collision bifurcations. In contrast to the system considered in the previous chapter, this system is unforced and piecewise-linear between switching events. We extend the symbolic representation developed in the previous chapter to study this system through bifurcations of Poincaré maps, enabling us to analytically derive border-collision bifurcations, while classical bifurcati ons are characterised numerically. \begin_inset ERT status open \begin_layout Plain Layout \backslash newline \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash noindent \end_layout \end_inset \series bold Chapter \begin_inset CommandInset ref LatexCommand ref reference "chap:Dynamics-of-targeted" plural "false" caps "false" noprefix "false" \end_inset \series default is largely reproduced from P. Ryan, J. Fokker, S. Healy and A. Amann, \emph on Dynamics of targeted ransomware negotiation \emph default , IEEE Access, 10:3283632844, 2022. \begin_inset CommandInset citation LatexCommand cite key "ryan2022dynamics" literal "false" \end_inset . Using game theory, we construct a stochastic model of negotiations between operators of targeted ransomware and their targets to better understand how to respond to ransomware attacks. In particular, our model considers the investments that a cybercriminal must make in order to conduct a successful targeted ransomware attack. We demonstrate how imperfect information is a crucial feature for replicating observed real-world behaviour. Furthermore, we present optimal strategies for both the perpetrator and the target, and demonstrate how imperfect information results in a non-trivial optimal strategy for the cybercriminal. This work has been found be of use in more recent research as the mathematical analysis of cybersecurity becomes more widely studied \begin_inset CommandInset citation LatexCommand cite key "guy2022indirect,parvaneh2021identify,skeochmodelling,bajpai2023know,teichmann2023evolution" literal "false" \end_inset . \begin_inset ERT status open \begin_layout Plain Layout \backslash newline \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash noindent \end_layout \end_inset \series bold Chapter \begin_inset CommandInset ref LatexCommand ref reference "chap:Parameter-estimation-and" plural "false" caps "false" noprefix "false" \end_inset \series default presents unpublished material in which we consider the problem of estimating and improving the efficiency of the data networks commonly used by cybersecurit y providers to deliver information to their customers. We develop a stochastic model of \begin_inset Quotes bld \end_inset Time-To-Live' data caching systems commonly used in cybersecurity with reference to real-world data derived from a cybersecurity data network. The Time-To-Live mechanism induces a time-delay effect in the system. We use our model to estimate metrics for caching system activity based on observable data, and to estimate the optimal Time-To-Live. The deterministic limit of the stochastic model gives rise to a circle map with periodic solutions that are analysed in terms of a symbolic representa tion. \end_layout \begin_layout Chapter Border-collision bifurcations in a driven time-delay system \begin_inset CommandInset label LatexCommand label name "chap:Border-collision-bifurcations-in" \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash renewcommand{ \backslash leftmark}{Border-collision bifurcations in a driven time-delay system} \end_layout \end_inset \end_layout \begin_layout Standard The material in this chapter is largely reproduced from P. Ryan, A. Keane and A. Amann, \emph on Border-collision bifurcations in a driven time-delay system \emph default , Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(2):023121, 2020. \begin_inset CommandInset citation LatexCommand cite key "ryan2020border" literal "false" \end_inset . My contribution to this paper consisted of exploring the parameter space and using the symbolic representation, which was proposed by Andreas, to calculate the existence and stability of solutions to study the bifurcations of the system. \end_layout \begin_layout Section Abstract \end_layout \begin_layout Standard We show that a simple piecewise-linear system with time delay and periodic forcing gives rise to a rich bifurcation structure of torus bifurcations and Arnold tongues, as well as multistability across a significant portion of the parameter space. The simplicity of our model enables us to study the dynamical features analytically. Specifically, these features are explained in terms of border collision bifurcations of an associated Poincaré map. Given that time delay and periodic forcing are common ingredients in mathematic al models, this analysis provides widely applicable insight. \end_layout \begin_layout Section Summary \end_layout \begin_layout Standard Both time delay dynamical systems, and periodically driven dynamical systems have been thoroughly studied in the literature. This can be attributed to their great relevance to real-world problems. Time delay systems arise naturally in physical, biological or climate models due to finite propagation speed; periodic drive is ubiquitous in engineering applications and is known to generate complex resonance phenomena. However, systems that combine these two properties have received much less attention, despite being relevant in many real-world applications. In this paper, we study a simple piecewise-linear system with both time delay and periodic forcing, which exhibits interesting dynamical features as a nontrivial consequence of this combination. These features include multistabilities, Arnold tongues, and torus bifurcations. Since the system is piecewise linear and contains only two parameters, many phenomena can be interpreted through an analytically derived piecewise-smo oth Poincaré map and an analysis of the associated border collision bifurcations. The analysis explains the origin of similar phenomena which has previously been observed numerically in more complicated related systems. \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard Periodically driven systems appear in many real-world applications. Examples include optical injection in laser systems \begin_inset CommandInset citation LatexCommand citep key "wieczorek_dynamical_2005" literal "true" \end_inset , vibration-driven energy harvesting devices \begin_inset CommandInset citation LatexCommand citep key "beeby_micro_2007" literal "true" \end_inset , injection-locked frequency dividers in electronics \begin_inset CommandInset citation LatexCommand citep key "daneshgar_observations_2010" literal "true" \end_inset , or seasonal forcing in climate systems \begin_inset CommandInset citation LatexCommand citep key "tziperman_nino_1994" literal "true" \end_inset . They often give rise to interesting resonance behaviour in damped oscillators \begin_inset CommandInset citation LatexCommand citep key "parlitz_superstructure_1985" literal "true" \end_inset and complex synchronization patterns in self-sustained oscillators \begin_inset CommandInset citation LatexCommand citep key "boccaletti_synchronization_2018,marchionne_synchronisation_2018" literal "true" \end_inset . \end_layout \begin_layout Standard Similarly, time delay systems also arise in many experimental systems, for example in optics \begin_inset CommandInset citation LatexCommand citep key "heil_chaos_2001,terrien2021pulse" literal "true" \end_inset , electronics \begin_inset CommandInset citation LatexCommand citep key "larger_virtual_2013" literal "true" \end_inset , neuro-science \begin_inset CommandInset citation LatexCommand citep key "scholl_time_2009" literal "true" \end_inset , or climate systems \begin_inset CommandInset citation LatexCommand citep key "runge_quantifying_2014" literal "true" \end_inset , and also play an important role in chaos control, for example, through the use of time delayed feedback control \begin_inset CommandInset citation LatexCommand citep key "pyragas_control_1995" literal "true" \end_inset . From a mathematical point of view, time delay often leads to a formally infinite-dimensional phase space \begin_inset CommandInset citation LatexCommand citep key "hale_introduction_2013" literal "true" \end_inset , which considerably complicates the analysis, but allows for a rich variety of phenomena. \end_layout \begin_layout Standard The combination of external forcing and time delay has been studied, for example, in the context of the Duffing oscillator \begin_inset CommandInset citation LatexCommand citep key "hu_resonances_1998" literal "true" \end_inset , the van der Pol oscillator \begin_inset CommandInset citation LatexCommand citep key "maccari_vibration_2003" literal "true" \end_inset and more recently in the context of climate systems \begin_inset CommandInset citation LatexCommand citep key "ghil_delay_2008,keane_delayed_2015" literal "true" \end_inset . However, a general understanding of this class of dynamical systems is not yet available. Here, our objective is to study the fundamental features of an elementary system with time delay and periodic forcing to obtain a broader insight into what phenomena are expected to arise as a consequence of this combination. \end_layout \begin_layout Standard Let us consider a simple driven time-delay dynamical system introduced by Ghil et al. \begin_inset CommandInset citation LatexCommand citep key "ghil_delay_2008" literal "true" \end_inset as a model for a climate phenomenon known as the El Niño Southern Oscillation. The system of a real variable \begin_inset Formula $x\in\mathbb{R}$ \end_inset is defined by \begin_inset Formula \begin{align} \dot{x}(t) & =-\text{\tanh}\left[\kappa x\left(t-\tau\right)\right]+b\sin\left(2\pi t\right)\label{eq:1}\\ x(t) & \in C\left(\left[-\tau,0\right]\right)\label{eq:2} \end{align} \end_inset where \begin_inset Formula $b\geq0$ \end_inset is the magnitude of the periodic forcing, \begin_inset Formula $\tau>0$ \end_inset is the time delay of the delayed feedback, and \begin_inset Formula $\kappa>0$ \end_inset is the linear slope of the delayed feedback at the origin. A solution of the system is a trajectory \begin_inset Formula $x(t)\in\mathbb{R}$ \end_inset , \begin_inset Formula $t\in\mathbb{R}$ \end_inset . A consequence of the reliance of the delayed feedback on a continuous function \begin_inset Formula $x\left(t\right)$ \end_inset over an interval \begin_inset Formula $\left[-\tau,0\right]$ \end_inset is that the system has an infinite-dimensional phase-space. Another key feature of this system is that it has the symmetry \begin_inset Formula $x\left(t\right)\rightarrow-x\left(t+\frac{1}{2}\right)$ \end_inset . This model has been studied extensively by Keane et al. \begin_inset CommandInset citation LatexCommand citep key "keane_delayed_2015,keane_investigating_2016" literal "true" \end_inset . Numerical analysis of this system demonstrated an extremely complex resonance structure \begin_inset CommandInset citation LatexCommand citep key "keane_delayed_2015" literal "true" \end_inset . Furthermore, the autonomous system \begin_inset Formula $\left(b=0\right)$ \end_inset has been studied analytically \begin_inset CommandInset citation LatexCommand citep key "nussbaum_uniqueness_1979,chow_characteristic_1988" literal "true" \end_inset . In this case the trivial solution \begin_inset Formula $x\equiv0$ \end_inset is only stable for \begin_inset Formula $\tau<\frac{\pi}{2\kappa}$ \end_inset . At \begin_inset Formula $\tau=\frac{\pi}{2\kappa}$ \end_inset , it becomes unstable, and a family of stable \begin_inset Formula $4\tau$ \end_inset -periodic solutions is born. \end_layout \begin_layout Standard In order to analyse the phenomena seen in this model further, let us consider a further simplification of the system by taking \begin_inset Formula $\kappa\rightarrow\infty$ \end_inset . This has the effect of changing the delayed feedback term from \begin_inset Formula $-\text{\tanh}\left[\kappa x\left(t-\tau\right)\right]$ \end_inset to \begin_inset Formula $-\text{\text{sgn}}\left[x\left(t-\tau\right)\right]$ \end_inset . We also apply the signum function to the periodic forcing to obtain the dynamical system \begin_inset Formula \begin{align} \dot{x}(t) & =-\text{sgn}\left[x\left(t-\tau\right)\right]+b\text{ sgn}\left(\sin\left(2\pi t\right)\right)\label{eq:3}\\ x(t) & \in C\left(\left[-\tau,0\right]\right)\label{eq:4} \end{align} \end_inset where \begin_inset Formula $b\geq0$ \end_inset and \begin_inset Formula $\tau>0$ \end_inset . Critically, this simplification of the system preserves the symmetry \begin_inset Formula $x\left(t\right)\rightarrow-x\left(t+\frac{1}{2}\right)$ \end_inset . The feedback term \begin_inset Formula $-\text{ sgn}\left[x\left(t-\tau\right)\right]$ \end_inset takes values in \begin_inset Formula $\left\{ 1,0,-1\right\} $ \end_inset . However, while the feedback can in principle be \begin_inset Formula $0$ \end_inset , this occurs only under highly specific conditions which are not considered here. The forcing term \begin_inset Formula $b\text{ sgn}\left[\sin\left(2\pi t\right)\right]$ \end_inset takes values in \begin_inset Formula $\left\{ b,0,-b\right\} $ \end_inset . As the forcing is \begin_inset Formula $0$ \end_inset only at discrete times, we say that the forcing is positive for \begin_inset Formula $t\bmod1\in[0,0.5)$ \end_inset , and negative for \begin_inset Formula $t\bmod1\in[0.5,1)$ \end_inset . We now develop our method for solving the system. \end_layout \begin_layout Section Numerics \end_layout \begin_layout Subsection Iterative map \end_layout \begin_layout Standard In order to solve Eqs. \begin_inset space ~ \end_inset ( \begin_inset CommandInset ref LatexCommand ref reference "eq:3" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "eq:4" \end_inset ), we note that \begin_inset Formula $\dot{x}(t)$ \end_inset can only take discrete values in \begin_inset Formula $\left\{ 1+b,-1+b,1-b,-1-b\right\} $ \end_inset ; therefore this continuous system can be modelled exactly as a discrete time iterative map, or Poincaré map. The state of the system at time \begin_inset Formula $t$ \end_inset is a tuple of variable length \begin_inset Formula \begin{equation} S(t)=\left(x;z_{0},z_{1},...,z_{n-1}\right) \end{equation} \end_inset where \begin_inset Formula $x\in\mathbb{R}$ \end_inset is the position at time \begin_inset Formula $t$ \end_inset and \begin_inset Formula $t-\tau0$ \end_inset in one cycle is shown in the top left corner of each plot. The vertical dotted lines indicate times when the forcing changes. \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard There are periodic and aperiodic solutions present in the system. A periodic solution with period \begin_inset Formula $P$ \end_inset is a trajectory that follows a cycle such that \begin_inset Formula $S(t+P)=S(t)$ \end_inset . An aperiodic solution is considered to be a solution with infinite period. We label solutions by the \emph on characteristic ratio \emph default \begin_inset Formula $P\mathbin{:}R$ \end_inset , where \begin_inset Formula $R$ \end_inset is the number of times the trajectory crosses from \begin_inset Formula $x<0$ \end_inset to \begin_inset Formula $x>0$ \end_inset in one cycle. Note that the characteristic ratio \begin_inset Formula $P\mathbin{:}R$ \end_inset is not the same as the frequency ratio \begin_inset Formula $p\mathbin{:}q$ \end_inset used in some previous literature \begin_inset CommandInset citation LatexCommand citep key "keane_delayed_2015,keane_investigating_2016" literal "true" \end_inset , where \begin_inset Formula $\frac{p}{q}$ \end_inset is the rotation number. The frequency ratio is less useful in this system because we do not observe any \begin_inset Formula $p:q$ \end_inset solutions where \begin_inset Formula $p\neq1$ \end_inset . The characteristic ratio \begin_inset Formula $P\mathbin{:}R$ \end_inset is a more descriptive measure of solutions in this system, and we will make extensive use of it in our analysis. \end_layout \begin_layout Standard Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (a) is an example of the stable solution to the unforced system \begin_inset Formula $\left(b=0\right)$ \end_inset . The change in the feedback occurs at a time \begin_inset Formula $t+\tau$ \end_inset after the trajectory passes through \begin_inset Formula $x=0$ \end_inset at time \begin_inset Formula $t$ \end_inset , resulting in a \begin_inset Formula $4\tau$ \end_inset -periodic solution that is stable for \begin_inset Formula $\tau>0$ \end_inset . This is consistent with the analytic results found for the unsimplified system. We consider \begin_inset Formula $4\tau$ \end_inset to be the natural period of the feedback, as the solution to the unforced system is \begin_inset Formula $4\tau$ \end_inset -periodic. The characteristic ratio of this solution is \begin_inset Formula $4\tau\mathbin{:}1$ \end_inset . \end_layout \begin_layout Standard Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (b) and Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (c) show a \begin_inset Formula $4\mathbin{:}2$ \end_inset solution and a \begin_inset Formula $3\mathbin{:}1$ \end_inset solution, respectively, that are stable for the same parameters. This is an example of bistability, where there are two stable solutions for the same parameters; the solution to which the system converges depends on which solution's basin of attraction the initial conditions are in. A second example of bistability is seen in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (d) and Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (e), which show a \begin_inset Formula $1\mathbin{:}1$ \end_inset solution and a \begin_inset Formula $5\mathbin{:}1$ \end_inset solution, respectively, that are stable for the same parameters. \end_layout \begin_layout Standard The solution in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (f) is assumed to be aperiodic, as the system does not converge to a periodic solution after running a simulation up to \begin_inset Formula $T=100000$ \end_inset . By comparing the aperiodic solution to the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (g), we may note that the aperiodic solution and the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution have the same number of \begin_inset Formula $x=0$ \end_inset crossings per period of the forcing. However, the \begin_inset Formula $x=0$ \end_inset crossings are evenly spaced in the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution, but are not in the aperiodic solution. This may be an indicator that the aperiodic solution is related to the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (g). A similar observation can be made for the \begin_inset Formula $3\mathbin{:}3$ \end_inset solution in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (h) and the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (d). In later sections of this chapter, we will investigate these relationships in detail. \end_layout \begin_layout Subsection Structure in the \begin_inset Formula $\left(b,\tau\right)$ \end_inset plane \end_layout \begin_layout Standard Having seen some interesting features of the system, we move on to understanding the overall dynamics in the \begin_inset Formula $(b,\tau)$ \end_inset plane. Due to the bistability present in the system, simulating the system from arbitrary initial conditions across a \begin_inset Formula $(b,\tau)$ \end_inset mesh would produce an inconsistent picture, as we have no prior knowledge of the basins of attraction of bistable solutions. In order to circumvent this issue, for fixed \begin_inset Formula $\tau$ \end_inset , the solution is swept across a range of \begin_inset Formula $b$ \end_inset by iteratively simulating the system, incrementing \begin_inset Formula $b$ \end_inset slightly, then simulating again using the final state of the previous simulatio n as the initial state of the next one. This allows a stable solution to be followed until it loses stability or ceases to exist, at which point the system converges to a nearby stable solution. By taking multiple sweeps in \begin_inset Formula $b$ \end_inset for a range of fixed \begin_inset Formula $\tau$ \end_inset values, we obtain the \begin_inset Formula $P\mathbin{:}R$ \end_inset charts shown in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset . Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset (a) shows the \begin_inset Formula $P\mathbin{:}R$ \end_inset chart under an upward sweep in \begin_inset Formula $b$ \end_inset , from left to right. Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset (b) shows the \begin_inset Formula $P\mathbin{:}R$ \end_inset chart under a downward sweep in \begin_inset Formula $b$ \end_inset , from right to left. This figure demonstrates many striking features of the system which will be explored in more detail. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../border_collision_bifurcations_in_a_driven_time_delay_system/fig_parameter_sweep_period.pdf width 85col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:period" \end_inset Colour maps showing the period of simulated solutions obtained by making 1124 sweeps of 1125 simulations of duration \begin_inset Formula $T=10000$ \end_inset in \begin_inset Formula $[0,8]\times[0,2.25]$ \end_inset . The white arrows indicate the direction of the sweep. The colour gradient shows the period \begin_inset Formula $P$ \end_inset of the observed solution on a log scale; white space indicates where the solution was aperiodic, or with \begin_inset Formula $P>999$ \end_inset . The white text indicates the characteristic ratio \begin_inset Formula $P:R$ \end_inset of stable solutions found within the labelled tongues. The dashed white line shows the border between the regions where the \begin_inset Formula $3\mathbin{:}1$ \end_inset and \begin_inset Formula $3\mathbin{:}3$ \end_inset solutions occur. \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard First, let us focus our attention on Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset (a), where we sweep from \series bold \begin_inset Formula $b=0$ \end_inset \series default up to \begin_inset Formula $b=8$ \end_inset . We observe that there are regions in the \begin_inset Formula $\left(b,\tau\right)$ \end_inset plane in which particular \begin_inset Formula $P\mathbin{:}R$ \end_inset solutions exist, such as the labelled \begin_inset Formula $3\mathbin{:}1$ \end_inset , \begin_inset Formula $5\mathbin{:}1$ \end_inset , \begin_inset Formula $7\mathbin{:}1$ \end_inset and \begin_inset Formula $9\mathbin{:}1$ \end_inset regions on the left side of the chart. These regions are sections of \emph on Arnold tongues \emph default . An Arnold tongue is a region of the \begin_inset Formula $\left(b,\tau\right)$ \end_inset plane, rooted on \begin_inset Formula $b=0$ \end_inset , within which the feedback and the forcing synchronise to produce a solution with period equal to a rational ratio of the forcing (and hence constant rotation number \begin_inset Formula $\frac{p}{q}$ \end_inset ). In piecewise-linear systems, an Arnold tongue can have shrinking points, at which the Arnold tongue has zero width. This phenomenon was first observed in the circle map \begin_inset CommandInset citation LatexCommand citep key "wei_arnold_1987" literal "true" \end_inset , and analysed in detail in the context of piecewise-linear continuous maps with single switching mechanisms \begin_inset CommandInset citation LatexCommand citep key "simpson_shrinking_2009,simpson_structure_2016,simpson_structure_2018" literal "true" \end_inset . Such Arnold tongues have been compared to strings of sausages \begin_inset CommandInset citation LatexCommand citep key "wei_arnold_1987" literal "true" \end_inset . We will refer to an individual \begin_inset Quotes eld \end_inset sausage \begin_inset Quotes erd \end_inset as a \emph on tongue \emph default , and refer to a full \begin_inset Quotes eld \end_inset string \begin_inset Quotes erd \end_inset as an Arnold tongue. A notable feature of the Arnold tongues in this system is that the characterist ic ratio is different in each tongue in the string. For example, the \begin_inset Formula $P=7$ \end_inset Arnold tongue is rooted on \begin_inset Formula $b=0$ \end_inset at \begin_inset Formula $\tau=1.75$ \end_inset , and consists of the \begin_inset Formula $7\mathbin{:}1$ \end_inset tongue, the \begin_inset Formula $7\mathbin{:}3$ \end_inset tongue, the \begin_inset Formula $7\mathbin{:}5$ \end_inset tongue, and the unlabelled \begin_inset Formula $7\mathbin{:}7$ \end_inset tongue. In each Arnold tongue, the leftmost tongue is a \begin_inset Formula $P\mathbin{:}R$ \end_inset tongue that is rooted on \begin_inset Formula $b=0$ \end_inset at \begin_inset Formula $\tau=\frac{P}{4R}$ \end_inset . \begin_inset Formula $P$ \end_inset is the same in every tongue in the chain, but \begin_inset Formula $R$ \end_inset increases the further right the tongue lies in the string. \end_layout \begin_layout Standard Tongues with similar characteristic ratios tend to have similar shape, with some variation. For example, the \begin_inset Formula $6\mathbin{:}2$ \end_inset and \begin_inset Formula $8\mathbin{:}2$ \end_inset tongues have identical shape; the \begin_inset Formula $4\mathbin{:}2$ \end_inset tongue is different. The large \begin_inset Formula $P\mathbin{:}1$ \end_inset tongues noted earlier have identical shape for \begin_inset Formula $P>1$ \end_inset . The \begin_inset Formula $1\mathbin{:}1$ \end_inset Arnold tongue is unlike any other Arnold tongue in shape. For large \begin_inset Formula $b$ \end_inset , the forcing dominates the feedback, which results in the system converging to the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution exclusively. We refer to the region in which \begin_inset Formula $b$ \end_inset is large enough as the locked region, where the system is locked to the period of the forcing. The boundary of the locked region is unusual, being made up of straight lines which meet at right angles. Branching off from the horizontal lines of the boundary, there are vertical stripes in which \begin_inset Formula $P\mathbin{:}P$ \end_inset solutions exist. We will devote considerable attention to studying the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution later. \end_layout \begin_layout Standard Now we compare the upward \begin_inset Formula $b$ \end_inset sweep in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset (a) to the downward \begin_inset Formula $b$ \end_inset sweep in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset (b). The most significant difference occurs around \begin_inset Formula $\left(b,\tau\right)=\left(2.5,1.25\right)$ \end_inset . In the upward sweep, the system follows the \begin_inset Formula $5\mathbin{:}1$ \end_inset solution to the edge of the \begin_inset Formula $5\mathbin{:}1$ \end_inset tongue, and there is a complicated region of smaller tongues and aperiodicity above the \begin_inset Formula $5\mathbin{:}1$ \end_inset tongue. In the downward sweep, the system instead continues to follow the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution into the region where different features occurred in the upward sweep. This agrees with the bistability of the \begin_inset Formula $1\mathbin{:}1$ \end_inset and \begin_inset Formula $5\mathbin{:}1$ \end_inset solutions seen in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (d,e). A less obvious difference is the bistability to the right of \begin_inset Formula $b=1$ \end_inset ; note the apparent difference in shape of the \begin_inset Formula $P\mathbin{:}2$ \end_inset and \begin_inset Formula $P\mathbin{:}1$ \end_inset tongues near this line. This occurs because the \begin_inset Formula $P\mathbin{:}1$ \end_inset tongues overlap with the \begin_inset Formula $P\mathbin{:}2$ \end_inset tongues, in agreement with the bistability of the \begin_inset Formula $4\mathbin{:}2$ \end_inset and \begin_inset Formula $3\mathbin{:}1$ \end_inset solutions seen in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (b,c). The absence of \begin_inset Formula $P\mathbin{:}1$ \end_inset tongues for even \begin_inset Formula $P$ \end_inset is notable. We numerically observe such solutions in simulations, but only for small \begin_inset Formula $b\apprle0.5$ \end_inset and exactly \begin_inset Formula $\tau=\frac{P}{4}$ \end_inset . \end_layout \begin_layout Section Dynamics \end_layout \begin_layout Standard Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset shows maximum charts in the \begin_inset Formula $\left(b,\tau\right)$ \end_inset plane overlayed with bifurcation curves. The maximum is taken as the maximum value of a solution over an interval of length \begin_inset Formula $1000$ \end_inset after a transient of length \begin_inset Formula $9000$ \end_inset from the same simulations that were used to generate Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset . Some of the larger tongues seen in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset can be seen without difficulty in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset due to a large jump in maxima at the boundaries of the tongues. We note that the even \begin_inset Formula $P\mathbin{:}R$ \end_inset tongues are striped horizontally; this is most evident in the \begin_inset Formula $4\mathbin{:}2$ \end_inset and \begin_inset Formula $8\mathbin{:}2$ \end_inset tongues. It appears that solutions with even \begin_inset Formula $P$ \end_inset or \begin_inset Formula $R$ \end_inset are not invariant under the symmetry \begin_inset Formula $x\left(t\right)\rightarrow-x\left(t+\frac{1}{2}\right)$ \end_inset ; rather there exists a pair of symmetry-related counterpart solutions, each with a different maximum value. The system converges to one of these solutions depending on initial conditions, and remains at that solution until swept out of the tongue, resulting in stripes parallel to the direction of the sweep. This feature was also observed in the smooth system ( \begin_inset CommandInset ref LatexCommand ref reference "eq:1" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "eq:2" \end_inset ) by \begin_inset CommandInset citation LatexCommand citet key "keane_delayed_2015" literal "true" \end_inset . \end_layout \begin_layout Standard We will devote the rest of this section to deriving and characterising the bifurcation curves plotted in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset . In order to do so, we first need to establish a systematic method of analysing solutions. We apply this method to develop Poincaré maps and border collision maps through which we study the bifurcations present in this system. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../border_collision_bifurcations_in_a_driven_time_delay_system/fig_parameter_sweep_bifurcations.pdf width 85col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:bifurcations" \end_inset Maximum charts in the \begin_inset Formula $(b,\tau)$ \end_inset plane overlayed with bifurcation curves. The colour gradient indicates the maximum value of the observed solution. The black arrows indicate the direction in which solutions were swept. The black text indicates the ratio of the period of the solution to the number of \begin_inset Formula $Z$ \end_inset symbols in the sequence for the stable solution within the associated tongue. BCSN bifurcations are shown in solid white, and T bifurcations are shown in solid black. The \begin_inset Formula $D\bar{D}$ \end_inset curve is shown in dashed white. \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Symbolic representation \end_layout \begin_layout Standard We require a robust framework under which we can analyse the dynamics of this system. We note that the characteristic ratio does not distinguish between the two different forms of the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution shown in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (d,g). Observing the order in which the feedback and forcing change after the trajectory passes through \begin_inset Formula $x=0$ \end_inset , we note that the feedback changes before the forcing in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (d), and after the forcing in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (g). In order to precisely capture these differences, a more explicit labelling system is required. We therefore adopt a symbolic representation of solutions. Symbolic representations have previously been used to great effect in the study of iterative maps \begin_inset CommandInset citation LatexCommand citep key "simpson_structure_2016,metropolis_finite_1973" literal "true" \end_inset . \end_layout \begin_layout Standard Let a solution be represented by a sequence of events \begin_inset Formula $...X_{1}X_{2}...X_{n}...$ \end_inset where \begin_inset Formula $X_{i}\in\left\{ D,\bar{D},Z,\bar{Z},H,\bar{H}\right\} .$ \end_inset \begin_inset Formula $D$ \end_inset denotes a transition of the forcing from \begin_inset Formula $-b$ \end_inset to \begin_inset Formula $b$ \end_inset , \begin_inset Formula $Z$ \end_inset denotes a transition from \begin_inset Formula $x<0$ \end_inset to \begin_inset Formula $x>0$ \end_inset , and \begin_inset Formula $H$ \end_inset denotes a transition of the feedback from \begin_inset Formula $-1$ \end_inset to \begin_inset Formula $1$ \end_inset . A bar over a symbol causes it to denote the opposite transition; a symbol with two bars over it is the same as the symbol unbarred. Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic" \end_inset (a) shows the \begin_inset Formula $5\mathbin{:}1$ \end_inset solution seen Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (e), labelled with the events that occur in the trajectory. As this solution is periodic, the sequence repeats, so we abbreviate the sequence of events representing the solution to a minimal repeating sequence \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D},D,\bar{D},\bar{Z},D,\bar{D},H,D,\bar{D},D\right]$ \end_inset . In general, a \begin_inset Formula $P\mathbin{:}R$ \end_inset solution is represented by a minimal repeating sequence of events \begin_inset Formula $\left[X_{1},X_{2},...,X_{n}\right]$ \end_inset containing: \end_layout \begin_layout Itemize \begin_inset Formula $P$ \end_inset \begin_inset Formula $D$ \end_inset events and \begin_inset Formula $P$ \end_inset \begin_inset Formula $\bar{D}$ \end_inset events, \end_layout \begin_layout Itemize \begin_inset Formula $R$ \end_inset \begin_inset Formula $Z$ \end_inset events and \begin_inset Formula $R$ \end_inset \begin_inset Formula $\bar{H}$ \end_inset events, \end_layout \begin_layout Itemize \begin_inset Formula $R$ \end_inset \begin_inset Formula $\bar{Z}$ \end_inset events and \begin_inset Formula $R$ \end_inset \begin_inset Formula $H$ \end_inset events. \end_layout \begin_layout Standard Every cyclic permutation of a sequence represents the same solution. For the sake of consistency, all sequences begins with a \begin_inset Formula $Z$ \end_inset . We observe that the \begin_inset Formula $5\mathbin{:}1$ \end_inset solution shown in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic" \end_inset (a) is invariant under the symmetry \begin_inset Formula $x\left(t\right)\rightarrow-x\left(t+\frac{1}{2}\right)$ \end_inset ; this causes the second half of the sequence \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D},D,\bar{D},\bar{Z},D,\bar{D},H,D,\bar{D},D\right]$ \end_inset to be the same as the first half with all symbols barred. We abbreviate \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D},D,\bar{D},\bar{Z},D,\bar{D},H,D,\bar{D},D\right]$ \end_inset as \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D},D,\bar{D}\right]^{-}$ \end_inset for convenience. In general, a \begin_inset Formula $P\mathbin{:}R$ \end_inset solution of the form \begin_inset Formula $\left[X_{1},X_{2},...X_{n},\bar{X_{1}},\bar{X_{2}},...,\bar{X_{n}}\right]$ \end_inset can also be represented by a half-sequence \begin_inset Formula $\left[X_{1},X_{2},...X_{n}\right]^{-}$ \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../border_collision_bifurcations_in_a_driven_time_delay_system/symbolic_solution.pdf width 85col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:symbolic" \end_inset Derivation of the symbolic representations of the \begin_inset Formula $5\mathbin{:}1$ \end_inset solution. \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard A sequence is considered \emph on legal \emph default within a subset of the \begin_inset Formula $(b,\tau)$ \end_inset plane if it represents a solution that exists within that subset. Each \begin_inset Formula $P\mathbin{:}R$ \end_inset solution is represented by a set of legal sequences, each one existing in a unique subset of the \begin_inset Formula $(b,\tau)$ \end_inset plane. The union of these subsets is the region in which the solution exists. The \begin_inset Formula $5\mathbin{:}1$ \end_inset solution exists within the \begin_inset Formula $5\mathbin{:}1$ \end_inset tongue shown in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset (a). It is represented by the half-sequences \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D},D,\bar{D}\right]^{-}$ \end_inset for \begin_inset Formula $\tau\leq1.25$ \end_inset and \begin_inset Formula $\left[Z,\bar{D},D,\bar{D},\bar{H},D,\bar{D}\right]^{-}$ \end_inset for \begin_inset Formula $\tau\geq1.25$ \end_inset , as shown in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic" \end_inset . At \begin_inset Formula $\tau=1.25$ \end_inset , the increasing time delay between \begin_inset Formula $Z$ \end_inset and \begin_inset Formula $\bar{H}$ \end_inset causes \begin_inset Formula $\bar{H}$ \end_inset to swap with \begin_inset Formula $\bar{D}$ \end_inset , changing the sequence. Determining whether a given sequence is legal within a subset of the \begin_inset Formula $(b,\tau)$ \end_inset plane is a nontrivial problem. Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Determining-legality-of" plural "false" caps "false" noprefix "false" \end_inset presents a general method to determine whether a sequence is legal for a given \begin_inset Formula $\left(b,\tau\right)$ \end_inset , which can also be used to calculate a solution analytically from a sequence for a given \begin_inset Formula $\left(b,\tau\right)$ \end_inset . This method was used to plot the solutions in Figs. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "fig:sn_bif" \end_inset and the bifurcation curves in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset . \end_layout \begin_layout Subsection Dimension of a periodic solution \begin_inset CommandInset label LatexCommand label name "subsec:Dimension-of-a" \end_inset \end_layout \begin_layout Standard The rest of our analysis will primarily be concerned with the stability of periodic solutions. However, before we can analyse the dynamics of this system, we must understand the phase space in which these dynamics take place. Time-delay systems are generically of uncountably infinite dimension, with the phase space being a continuous function over a finite interval. Under certain circumstances, the effective dimension of smooth time-delay systems can be reduced through dimension reduction techniques \begin_inset CommandInset citation LatexCommand cite key "stepan2006stability,insperger2010dimension,wang2012seasonal,nandakumar2013galerkin" literal "false" \end_inset . To avoid confusion, we note that such techniques are not being used here. Instead, we employ a somewhat novel approach inspired by our use of Poincaré maps to study this system. \end_layout \begin_layout Standard As discussed previously, the state of the system \begin_inset Formula $S\left(t\right)$ \end_inset is a tuple whose length varies dynamically as the number of zero crossings in the last tau time varies. Consider the \begin_inset Formula $5:1$ \end_inset solution represented by \begin_inset Formula $\left[Z,\bar{D},D,\bar{D},\bar{H},D,\bar{D}\right]^{-}$ \end_inset in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic" \end_inset (b). At any time after a \begin_inset Formula $\bar{H}$ \end_inset and before a \begin_inset Formula $\bar{Z}$ \end_inset , or after a \begin_inset Formula $H$ \end_inset and before a \begin_inset Formula $Z$ \end_inset , there are no history elements in the state, and so \begin_inset Formula $S(t)=\left(x;z_{0}\right)$ \end_inset . At any time after a \begin_inset Formula $Z$ \end_inset and before the associated \begin_inset Formula $\bar{H}$ \end_inset , or after a \begin_inset Formula $\bar{Z}$ \end_inset and before the associated \begin_inset Formula $H$ \end_inset , there is one history element in the state, and so \begin_inset Formula $S(t)=\left(x;z_{0}\right)$ \end_inset . However, by making use the sequence representation of the solution, if you know the displacement \begin_inset Formula $x$ \end_inset , you can calculate the history element \begin_inset Formula $z_{0}$ \end_inset , and so the state can be reduced to a single independent variable. To take a simple example, at time \begin_inset Formula $t=0$ \end_inset , a \begin_inset Formula $D$ \end_inset event takes place at \begin_inset Formula $x=x_{D}$ \end_inset , and the preceding \begin_inset Formula $Z$ \end_inset event takes place at \begin_inset Formula $x=0$ \end_inset at time \begin_inset Formula $t=z_{0}$ \end_inset . Then \begin_inset Formula \begin{equation} x_{D}=0+\left(-\frac{1}{2}-z_{0}\right)\left(1+b\right)+\frac{1}{2}\left(1-b\right) \end{equation} \end_inset which we can solve for \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset Formula $z_{0}$ \end_inset \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit to get \begin_inset Formula \begin{align} z_{0} & =-\frac{b+x_{D}}{b+1} \end{align} \end_inset We say then that the \begin_inset Formula $5:1$ \end_inset solution is one-dimensional, as the state consists of either a single dynamic variable, or two linearly dependent dynamic variables. \end_layout \begin_layout Paragraph Definition: \end_layout \begin_layout Standard The dimension of a periodic solution is the maximal number of independent dynamic variables in the state over a single cycle of the solution. \end_layout \begin_layout Standard This is equivalent to saying that the dimension of a periodic solution is the minimal number of dynamic required to construct a Poincaré map which captures the dynamics of that solution. A one-dimensional Poincaré map is sufficient to capture the stability and existence of the \begin_inset Formula $5:1$ \end_inset solution, as we will demonstrate later. However, to fully consider the dynamics of the \begin_inset Formula $5:1$ \end_inset solution we will require additional concepts introduced later. Instead, we will begin with the \begin_inset Formula $1:1$ \end_inset solution which dominates so much of the parameter space. \end_layout \begin_layout Subsection Torus bifurcation of the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution \begin_inset CommandInset label LatexCommand label name "subsec:1:1_solution" \end_inset \end_layout \begin_layout Standard We now apply our sequence representation in analysing the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution; specifically, we determine what happens at the vertical black lines along the boundary of the locked region in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset . Consider the set of half-sequences that represent the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution. Each half-sequence contains one \begin_inset Formula $Z$ \end_inset or \begin_inset Formula $\bar{Z}$ \end_inset , one \begin_inset Formula $H$ \end_inset or \begin_inset Formula $\bar{H}$ \end_inset , and one \begin_inset Formula $D$ \end_inset or \begin_inset Formula $\bar{D}$ \end_inset . We need only consider half-sequences starting with \begin_inset Formula $Z$ \end_inset , as \begin_inset Formula $\left[X_{1},X_{2},X_{3}\right]^{-}=\left[X_{3},X_{1},X_{2}\right]^{-}=\left[X_{2},X_{3},X_{1}\right]^{-}$ \end_inset . There are only eight possibilities: \begin_inset Formula $\left[Z,H,D\right]^{-}$ \end_inset , \begin_inset Formula $\left[Z,D,H\right]^{-}$ \end_inset , \begin_inset Formula $\left[Z,\bar{H},D\right]^{-}$ \end_inset , \begin_inset Formula $\left[Z,D,\bar{H}\right]^{-}$ \end_inset , \begin_inset Formula $\left[Z,H,\bar{D}\right]^{-}$ \end_inset , \begin_inset Formula $\left[Z,\bar{D},H\right]^{-}$ \end_inset , \begin_inset Formula $\left[Z,\bar{H},\bar{D}\right]^{-}$ \end_inset , \begin_inset Formula $\left[Z,\bar{D},\bar{H}\right]^{-}$ \end_inset . \begin_inset Formula $\left[Z,H,D\right]^{-}$ \end_inset and \begin_inset Formula $\left[Z,D,H\right]^{-}$ \end_inset are not legal, as they require \begin_inset Formula $Z$ \end_inset to occur when \begin_inset Formula $\dot{x}<0$ \end_inset , which is impossible. Similarly, \begin_inset Formula $\left[Z,\bar{H},D\right]^{-}$ \end_inset and \begin_inset Formula $\left[Z,D,\bar{H}\right]^{-}$ \end_inset are legal only for \begin_inset Formula $b<1$ \end_inset , and \begin_inset Formula $\left[Z,\bar{D},H\right]^{-}$ \end_inset and \begin_inset Formula $\left[Z,H,\bar{D}\right]^{-}$ \end_inset are legal only for \begin_inset Formula $b>1$ \end_inset . The \begin_inset Formula $1\mathbin{:}1$ \end_inset solutions shown in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset are \begin_inset Formula $\left[Z,\bar{H},\bar{D}\right]^{-}$ \end_inset in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (d) and \begin_inset Formula $\left[Z,\bar{D},\bar{H}\right]^{-}$ \end_inset in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sample_solutions" \end_inset (g). In the case \begin_inset Formula $b>1$ \end_inset , the sequence representing the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution is restricted by \begin_inset Formula $\tau$ \end_inset in the following way: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \begin{aligned}\left[Z,\bar{H},\bar{D},\bar{Z},H,D\right]\text{ for } & \tau\bmod1\in\left[0,0.25\right),\\ \left[Z,\bar{D},\bar{H},\bar{Z},D,H\right]\text{ for } & \tau\bmod1\in\left[0.25,0.5\right),\\ \left[Z,H,\bar{D},\bar{Z},\bar{H},D\right]\text{ for } & \tau\bmod1\in\left[0.5,0.75\right),\\ \left[Z,\bar{D},H,\bar{Z},D,\bar{H}\right]\text{ for } & \ensuremath{\tau\bmod1\in\left[0.75,1\right)}. \end{aligned} \label{eq:7} \end{equation} \end_inset This can be explained by observing that for small \begin_inset Formula $\tau$ \end_inset , \begin_inset Formula $\bar{H}$ \end_inset must follow the \begin_inset Formula $Z$ \end_inset that created it almost immediately. As \begin_inset Formula $\tau$ \end_inset increases, \begin_inset Formula $\bar{H}$ \end_inset drifts further away from \begin_inset Formula $Z$ \end_inset in the sequence, drifting past \begin_inset Formula $\bar{D}$ \end_inset at \begin_inset Formula $\tau=0.25$ \end_inset . At \begin_inset Formula $\tau=0.5$ \end_inset , \begin_inset Formula $\bar{H}$ \end_inset drifts past the subsequent \begin_inset Formula $\bar{Z}$ \end_inset . At \begin_inset Formula $\tau=0.75$ \end_inset , \begin_inset Formula $\bar{H}$ \end_inset drifts past the subsequent \begin_inset Formula $D$ \end_inset . At \begin_inset Formula $\tau=1$ \end_inset , the \begin_inset Formula $\bar{H}$ \end_inset created at \begin_inset Formula $Z$ \end_inset drifts past the subsequent \begin_inset Formula $Z$ \end_inset , and the pattern repeats. At the same time, the same drift pattern occurs between \begin_inset Formula $\bar{Z}$ \end_inset and \begin_inset Formula $H$ \end_inset . We now apply this information to analyse the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution. \end_layout \begin_layout Standard We construct a Poincaré map on the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution. Let \begin_inset Formula $t_{z}$ \end_inset be a time on the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution at which \begin_inset Formula $x=0$ \end_inset ; then the state of the system at time \begin_inset Formula $t_{z}$ \end_inset is \begin_inset Formula \begin{equation} S(t_{z})=\left(0;z_{0},z_{1},...,z_{n-2},t_{z}\right)^{T} \end{equation} \end_inset As \begin_inset Formula $x=0$ \end_inset at \begin_inset Formula $t=t_{z}$ \end_inset , we drop \begin_inset Formula $x$ \end_inset and rewrite the state as \begin_inset Formula \begin{equation} S_{z}=\left(z_{0},z_{1},...,z_{n-2},t_{z}\right)^{T} \end{equation} \end_inset Let \begin_inset Formula $t_{z}^{*}$ \end_inset be the time on the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution at which \begin_inset Formula $x=0$ \end_inset immediately after \begin_inset Formula $t_{z}$ \end_inset . Then the state of the system at time \begin_inset Formula $t_{z}^{*}$ \end_inset is \begin_inset Formula \begin{equation} S_{z}^{*}=\left(z_{1},z_{2},...,t_{z},t_{z}^{*}\right)^{T} \end{equation} \end_inset As the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution is invariant under the symmetry \begin_inset Formula $x\left(t\right)\rightarrow-x\left(t+\frac{1}{2}\right)$ \end_inset , it has the property \begin_inset Formula \begin{equation} S_{z}=\left(\begin{array}{c} z_{0}\\ z_{1}\\ ...\\ z_{n-2}\\ t_{z} \end{array}\right)=\left(\begin{array}{c} z_{1}-\frac{1}{2}\\ z_{2}-\frac{1}{2}\\ ...\\ t_{z}-\frac{1}{2}\\ t_{z}^{*}-\frac{1}{2} \end{array}\right)=S_{z}^{*}-\frac{1}{2}\label{eq:11} \end{equation} \end_inset We define a Poincaré map \begin_inset Formula $\mathbb{P}:S_{z}\rightarrow S_{z}^{*}-\frac{1}{2}$ \end_inset so that the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution is a fixed point of \begin_inset Formula $\mathbb{P}$ \end_inset . Therefore, \begin_inset Formula $\mathbb{P}$ \end_inset is defined by \begin_inset Formula \begin{equation} \mathbb{P}\left(\begin{array}{c} z_{0}\\ z_{1}\\ ...\\ z_{n-2}\\ t_{z} \end{array}\right)=\left(\begin{array}{c} z_{1}-\frac{1}{2}\\ z_{2}-\frac{1}{2}\\ ...\\ t_{z}-\frac{1}{2}\\ t_{z}^{*}-\frac{1}{2} \end{array}\right)\label{eq:14} \end{equation} \end_inset For the purpose of deriving \begin_inset Formula $\mathbb{P}$ \end_inset , we will assume w.l.o.g. that the trajectory is transitioning from \begin_inset Formula $x<0$ \end_inset to \begin_inset Formula $x>0$ \end_inset at \begin_inset Formula $t_{z}\in\left[0,0.5\right)$ \end_inset . Let \begin_inset Formula $t_{h}\in\left[t_{z},t_{z}^{*}\right)$ \end_inset be the time at which the feedback changes. First, we consider the case where \begin_inset Formula $\tau<0.5$ \end_inset ; then the zero element generated at \begin_inset Formula $t_{z}$ \end_inset is consumed at \begin_inset Formula $t_{h}=t_{z}+\tau$ \end_inset . Then \begin_inset Formula $\mathbb{P}$ \end_inset is one-dimensional as \begin_inset Formula $S_{z}$ \end_inset has only one variable, \begin_inset Formula $t_{z}$ \end_inset . We write the one-dimensional map as \begin_inset Formula \begin{equation} \mathbb{P}\left(t_{z}\right)=t_{z}^{*}-\frac{1}{2}\label{eq:13-1} \end{equation} \end_inset If \begin_inset Formula $\tau<0.5$ \end_inset , then the solution is represented by the sequence \begin_inset Formula $\left[Z,\bar{H},\bar{D}\right]^{-}$ \end_inset ; therefore, the feedback is positive for \begin_inset Formula $t\in[t_{z},t_{h})$ \end_inset . We write an equation for the displacement of the trajectory between \begin_inset Formula $t_{z}$ \end_inset and \begin_inset Formula $t_{z}^{*}$ \end_inset as \begin_inset Formula \begin{equation} b\left(\frac{1}{2}-t_{z}\right)-b\left(t_{z}^{*}-\frac{1}{2}\right)+(t_{h}-t_{z})-\left(t_{z}^{*}-t_{h}\right)=0\label{eq:9} \end{equation} \end_inset We substitute \begin_inset Formula $t_{h}=t_{z}+\tau$ \end_inset and solve for \begin_inset Formula $t_{z}^{*}-\frac{1}{2}$ \end_inset to obtain \begin_inset Formula \begin{equation} \mathbb{P}(t_{z})=-\left(\frac{b-1}{b+1}\right)t_{z}+\frac{b+2\tau}{b+1}-\frac{1}{2} \end{equation} \end_inset As \begin_inset Formula $\left|\frac{b-1}{b+1}\right|<1$ \end_inset for \begin_inset Formula $b\geq0$ \end_inset , \begin_inset Formula $\mathbb{P}$ \end_inset is stable for \begin_inset Formula $\tau<\frac{1}{2}$ \end_inset . \end_layout \begin_layout Standard If \begin_inset Formula $\tau\in\left[0.5,1\right)$ \end_inset , then \begin_inset Formula $t_{h}$ \end_inset was generated, not at \begin_inset Formula $t_{z}$ \end_inset , but at the previous \begin_inset Formula $x=0$ \end_inset crossing \begin_inset Formula $z_{0}$ \end_inset . Then \begin_inset Formula $\mathbb{P}$ \end_inset is two-dimensional as \begin_inset Formula $S_{z}$ \end_inset has two variables, and the feedback is negative for \begin_inset Formula $t\in\left[t_{z},t_{h}\right)$ \end_inset , where \begin_inset Formula $t_{h}=z_{0}+\tau$ \end_inset . These conditions can be generalised for larger \begin_inset Formula $\tau$ \end_inset . The dimension of the system is \begin_inset Formula $n=\left\lceil 2\tau\right\rceil $ \end_inset , where \begin_inset Formula $\left\lceil 2\tau\right\rceil $ \end_inset is the smallest integer greater than \begin_inset Formula $2\tau$ \end_inset ; then the feedback for \begin_inset Formula $t\in\left[t_{z},t_{h}\right)$ \end_inset is \begin_inset Formula $(-1)^{n-1}$ \end_inset . We can then generalise Eq. ( \begin_inset CommandInset ref LatexCommand eqref reference "eq:9" \end_inset ) for arbitrary \begin_inset Formula $n$ \end_inset and solve for \begin_inset Formula $t_{z}^{*}-\frac{1}{2}$ \end_inset to obtain \begin_inset Formula \begin{equation} t_{z}^{*}-\frac{1}{2}=-t_{z}+\frac{b+2t_{h}(-1)^{n-1}}{b+(-1)^{n-1}}-\frac{1}{2}\label{eq:13} \end{equation} \end_inset For \begin_inset Formula $\tau>\frac{1}{2}$ \end_inset , \begin_inset Formula $\mathbb{P}$ \end_inset can written as \begin_inset Formula \begin{equation} \mathbb{P}\left(S_{z}\right)=AS_{z}+B \end{equation} \end_inset where \begin_inset Formula $A$ \end_inset is an \begin_inset Formula $n\times n$ \end_inset matrix \begin_inset Formula \begin{equation} A=\left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0\\ \vdots & 0 & \ddots & \ddots & \vdots\\ \vdots & \vdots & \ddots & \ddots & 0\\ 0 & 0 & \cdots & 0 & 1\\ \frac{2(-1)^{n-1}}{b+(-1)^{n-1}} & 0 & \cdots & \cdots & -1 \end{array}\right) \end{equation} \end_inset and \begin_inset Formula \begin{equation} B=\left(\begin{array}{c} -\frac{1}{2}\\ -\frac{1}{2}\\ -\frac{1}{2}\\ -\frac{1}{2}\\ ...\\ -\frac{1}{2}\\ \frac{b+2\tau(-1)^{n-1}}{b+(-1)^{n-1}}-\frac{1}{2} \end{array}\right) \end{equation} \end_inset Note that \begin_inset Formula $A$ \end_inset only depends on the parameter \begin_inset Formula $b$ \end_inset and not on \begin_inset Formula $\tau$ \end_inset . This means that for fixed \begin_inset Formula $n$ \end_inset , the value of of \begin_inset Formula $b$ \end_inset at which the fixed point of \begin_inset Formula $\mathbb{P}$ \end_inset is bifurcating, \begin_inset Formula $b_{\text{bif}}$ \end_inset , is constant. By solving the characteristic equation, we calculate \begin_inset Formula \begin{equation} b_{\text{bif}}(n)=\frac{1}{\cos\left(\frac{\pi\left(n-1\right)}{2n-1}\right)}-(-1)^{n-1} \end{equation} \end_inset where \begin_inset Formula $n=\left\lceil 2\tau\right\rceil $ \end_inset . Full calculations may be found in Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Calculation-of-the" plural "false" caps "false" noprefix "false" \end_inset . The curve \begin_inset Formula $\left(b_{\text{bif}}\left(\left\lceil 2\tau\right\rceil \right),\tau\right)$ \end_inset can be seen plotted as black vertical lines against maximum charts in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset \emph on . \emph default If we examine the maximum chart where \begin_inset Formula $b$ \end_inset is swept from right to left, we see that the system ceases to converge to the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution along this curve. We now know that this is because the fixed point of \begin_inset Formula $\mathbb{P}$ \end_inset , and hence the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution, loses stability at \begin_inset Formula $b=b_{\text{bif}}$ \end_inset . By calculating the eigenvalues of \begin_inset Formula $\mathbb{P}$ \end_inset explicitly for \begin_inset Formula $n=2$ \end_inset and \begin_inset Formula $n=3$ \end_inset , we find that the loss of stability occurs because a pair of complex conjugate eigenvalues \begin_inset Formula $\lambda_{1,2}$ \end_inset cross \begin_inset Formula $|\lambda|=1$ \end_inset . Therefore the fixed point of \begin_inset Formula $\mathbb{P}$ \end_inset loses stability due to a Neimark-Sacker (NS) bifurcation \begin_inset CommandInset citation LatexCommand citep key "hone_neimarksacker_2010" literal "true" \end_inset . As \begin_inset Formula $\mathbb{P}$ \end_inset is a Poincaré map on a periodic orbit, a NS bifurcation of the fixed point of \begin_inset Formula $\mathbb{P}$ \end_inset corresponds to a torus (T) bifurcation of the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution. However, \begin_inset Formula $\mathbb{P}$ \end_inset only shows the existence of the T bifurcation along the vertical sections of the boundary of the locked region, where \begin_inset Formula $n$ \end_inset is constant and \begin_inset Formula $\mathbb{P}$ \end_inset is smooth. To fully understand the horizontal sections of the boundary, we must look to non-smooth bifurcation theory. \end_layout \begin_layout Section Border-collision bifurcations \end_layout \begin_layout Standard The Arnold tongues seen in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset have sharply defined boundaries, made up of curves and straight lines. We seek to determine what happens to solutions at these boundaries and derive analytic expressions for the boundaries using Poincaré maps. \end_layout \begin_layout Standard By simulating solutions near the boundaries of the Arnold tongues, we observe that moving closer to the boundaries causes a \begin_inset Formula $D$ \end_inset or \begin_inset Formula $\bar{D}$ \end_inset to move closer to \begin_inset Formula $x=0$ \end_inset . An example of this is Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic" \end_inset (a), where the \begin_inset Formula $D$ \end_inset at \begin_inset Formula $t=0$ \end_inset in the \begin_inset Formula $5\mathbin{:}1$ \end_inset solution occurs for \begin_inset Formula $x>0$ \end_inset near \begin_inset Formula $x=0$ \end_inset . If this \begin_inset Formula $D$ \end_inset crossed \begin_inset Formula $x=0$ \end_inset and occurred at \begin_inset Formula $x<0$ \end_inset , this would significantly impact the feedback. There would be two additional \begin_inset Formula $x=0$ \end_inset crossings in the trajectory, changing the \begin_inset Formula $D$ \end_inset to \begin_inset Formula $\bar{Z}DZ$ \end_inset and adding a \begin_inset Formula $H$ \end_inset and a \begin_inset Formula $\bar{H}$ \end_inset elsewhere in the sequence. Therefore, when we construct a Poincaré map to describe the dynamics of the system close to the boundary of such a tongue, the map must have a border at \begin_inset Formula $x=0$ \end_inset , such that the map is continuous across the border but not differentiable at the border. Such maps, and the associated border collision bifurcations, have recently received systematic analysis in the literature \begin_inset CommandInset citation LatexCommand citep key "bernardo_bifurcations_2002,banerjee_border_1999,colombo_bifurcations_2012,di_bernardo_bifurcations_2008,granados_border_2014,meiss_neimarksacker_2008,nusse_border-collision_1994,holmberg_relay_1993,colombo_complex_2007" literal "true" \end_inset . \end_layout \begin_layout Subsection Border-collision saddle-node bifurcation of the \begin_inset Formula $5\mathbin{:}1$ \end_inset and \begin_inset Formula $5\mathbin{:}3$ \end_inset solutions \end_layout \begin_layout Standard To begin our analysis of the border-collision bifurcations in this system, we consider what happens to the observed \begin_inset Formula $5:1$ \end_inset solution at the boundary of the \begin_inset Formula $5:1$ \end_inset tongue. We begin with this solution for two reasons. Firstly, we observe that for odd \begin_inset Formula $P\geq5$ \end_inset , the \begin_inset Formula $P:1$ \end_inset tongues are all of the same shape, and the \begin_inset Formula $P:1$ \end_inset solution vanishes at the boundary of the tongue. This is in contrast to the \begin_inset Formula $3:1$ \end_inset solution, which may change smoothly to the \begin_inset Formula $3:3$ \end_inset solution at the boundary, and the \begin_inset Formula $1:1$ \end_inset solution, which behaves unlike any other periodic solution. Secondly, as an odd period solution, it has the symmetry \begin_inset Formula $x(t)=-x(t+\frac{P}{2})$ \end_inset ; this makes the construction of Poincaré maps easier than it is for the even period solutions. \end_layout \begin_layout Standard We construct a Poincaré map \begin_inset Formula $\mathbb{B}$ \end_inset on the \begin_inset Formula $5\mathbin{:}1$ \end_inset and \begin_inset Formula $5\mathbin{:}3$ \end_inset solutions represented by \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D},D,\bar{D}\right]^{-}$ \end_inset and \begin_inset Formula $\left[Z,\bar{D},\bar{H},H,D,\bar{H},\bar{D},\bar{Z},D,Z,\bar{D}\right]^{-}$ \end_inset respectively, which can be seen in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sn_bif" \end_inset (a), such that our fixed point is the state of the system at time \begin_inset Formula $t=0$ \end_inset , at the position \begin_inset Formula $x_{D}$ \end_inset at which the \begin_inset Formula $D$ \end_inset that will cross \begin_inset Formula $x=0$ \end_inset occurs. As noted in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:Dimension-of-a" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset Formula $\mathbb{B}$ \end_inset is one-dimensional. We divide \begin_inset Formula $\mathbb{B}$ \end_inset into \begin_inset Formula $\mathbb{B}^{+}$ \end_inset and \begin_inset Formula $\mathbb{B}^{-}$ \end_inset for \begin_inset Formula $x\geq0$ \end_inset and \begin_inset Formula $x<0$ \end_inset respectively. The \begin_inset Formula $5\mathbin{:}1$ \end_inset and \begin_inset Formula $5\mathbin{:}3$ \end_inset solutions are invariant under the symmetry \begin_inset Formula $x(t)=-x\left(t+\frac{1}{2}\right)$ \end_inset ; therefore we construct our map \begin_inset Formula $\mathbb{B}:x_{D}\left(0\right)\rightarrow-x_{D}\left(\frac{5}{2}\right)$ \end_inset . \begin_inset Formula $\mathbb{B^{\mathrm{+}}}$ \end_inset is a map on the solution represented by \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D},D,\bar{D}\right]^{-}$ \end_inset . Following the blue curve in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sn_bif" \end_inset (b), we can derive \begin_inset Formula \begin{equation} \mathbb{B}^{+}(x_{D})=-\left(\frac{b-1}{b+1}\right)x_{D}+\frac{2b}{b+1}-\frac{b+5}{2}+2\tau \end{equation} \end_inset \begin_inset Formula $\mathbb{B}^{-}$ \end_inset is a map on the \begin_inset Formula $5\mathbin{:}3$ \end_inset solution represented by \begin_inset Formula $\left[Z,\bar{D},\bar{H},H,D,\bar{H},\bar{D},\bar{Z},D,Z,\bar{D}\right]^{-}$ \end_inset , which is shown in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sn_bif" \end_inset (b) in red. The feedback change due the trajectory dipping below \begin_inset Formula $x=0$ \end_inset occurs at some \begin_inset Formula $t\in\left(1,1.5\right)$ \end_inset has duration \begin_inset Formula $\frac{-2bx}{b^{2}-1}$ \end_inset ; \begin_inset Formula $\mathbb{B}^{-}$ \end_inset is otherwise identical to \begin_inset Formula $\mathbb{B^{\mathrm{+}}}$ \end_inset . Thus we derive \begin_inset Formula $\mathbb{B}$ \end_inset as \begin_inset Formula \begin{equation} \mathbb{B}(x_{D})=\begin{cases} -\left(\frac{b-1}{b+1}\right)x_{D}+\frac{2b}{b+1}-\frac{b+5}{2}+2\tau & x_{D}\geq0\\ \left(\frac{4b}{b^{2}-1}-\frac{b-1}{b+1}\right)x_{D}+\frac{2b}{b+1}-\frac{b+5}{2}+2\tau & x_{D}<0 \end{cases} \end{equation} \end_inset By setting \begin_inset Formula $x_{D}=0$ \end_inset and solving \begin_inset Formula $\mathbb{B}$ \end_inset for \begin_inset Formula $\tau$ \end_inset , we obtain the curve \begin_inset Formula $\tau=\frac{b^{2}+2b+5}{4\left(b+1\right)}$ \end_inset . This matches the lower right boundary of the \begin_inset Formula $5\mathbin{:}1$ \end_inset tongue spanning from \begin_inset Formula $\left(1,1\right)$ \end_inset to \begin_inset Formula $\left(3,1.25\right)$ \end_inset . For \begin_inset Formula $b\in\left[1,3\right]$ \end_inset and \begin_inset Formula $\tau\in\left[\frac{b^{2}+2b+5}{4\left(b+1\right)},1.25\right]$ \end_inset , \begin_inset Formula $\mathbb{B}$ \end_inset has two fixed points; a stable fixed point that exists for \begin_inset Formula $x_{D}\geq0$ \end_inset and an unstable fixed point that exists for \begin_inset Formula $x_{D}\leq0$ \end_inset . These two fixed points collide and vanish at the border \begin_inset Formula $x_{D}=0$ \end_inset at \begin_inset Formula $\tau=\frac{b^{2}+2b+5}{4\left(b+1\right)}$ \end_inset in a border collision saddle node (BCSN) bifurcation \begin_inset CommandInset citation LatexCommand citep key "banerjee_border_1999" literal "true" \end_inset . Hence the \begin_inset Formula $5\mathbin{:}1$ \end_inset tongue is bounded by a BCSN bifurcation. Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sn_bif" \end_inset (b,e,f) shows the BCSN bifurcation of the \begin_inset Formula $5\mathbin{:}1$ \end_inset and \begin_inset Formula $5\mathbin{:}3$ \end_inset solutions. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../border_collision_bifurcations_in_a_driven_time_delay_system/sn_bifurcating_solutions.pdf width 70col% \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../border_collision_bifurcations_in_a_driven_time_delay_system/sn_bifurcation_diagram.pdf width 75col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:sn_bif" \end_inset BCSN bifurcations at the boundary of the \begin_inset Formula $5\mathbin{:}1$ \end_inset tongue. The stable solution is plotted in solid blue, while the unstable solution is plotted in dashed red. In \begin_inset Formula $(a)$ \end_inset , the stable \begin_inset Formula $5\mathbin{:}1$ \end_inset solution \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D},D,\bar{D}\right]^{-}$ \end_inset and the unstable \begin_inset Formula $5\mathbin{:}1$ \end_inset solution \begin_inset Formula $\left[Z,D,\bar{D},D,\bar{H},\bar{D},D\right]^{-}$ \end_inset are plotted close to \begin_inset Formula $\left(b,\frac{5-b}{4}\right)$ \end_inset . In \begin_inset Formula $(b)$ \end_inset , the stable \begin_inset Formula $5\mathbin{:}1$ \end_inset solution \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D},D,\bar{D}\right]^{-}$ \end_inset and the unstable \begin_inset Formula $5\mathbin{:}3$ \end_inset solution \begin_inset Formula $\left[Z,\bar{D},\bar{H},H,D,\bar{H},\bar{D},\bar{Z},D,Z,\bar{D}\right]^{-}$ \end_inset are plotted close to \begin_inset Formula $\left(b,\frac{b^{2}+2b+5}{4\left(b+1\right)}\right)$ \end_inset . The \begin_inset Formula $D$ \end_inset events associated with the BCSN bifurcation for \begin_inset Formula $b<1$ \end_inset are plotted as triangles, while those associated with the BCSN bifurcation for \begin_inset Formula $b>1$ \end_inset are plotted as circles; \begin_inset Formula $(c)$ \end_inset , \begin_inset Formula $(d)$ \end_inset , \begin_inset Formula $(e)$ \end_inset and \begin_inset Formula $(f)$ \end_inset show their evolution as \begin_inset Formula $b$ \end_inset and \begin_inset Formula $\tau$ \end_inset vary. \end_layout \end_inset \end_layout \end_inset We generalise \begin_inset Formula $\mathbb{B}$ \end_inset for every \begin_inset Formula $P\mathbin{:}1$ \end_inset tongue for odd \begin_inset Formula $P\geq5$ \end_inset as \begin_inset Formula \begin{equation} \mathbb{B}_{P}(x_{D})=\begin{cases} -\left(\frac{b-1}{b+1}\right)x_{D}+\frac{2b}{b+1}-\frac{b+\left|P-4\tau\right|}{2} & x_{D}\geq0\\ \left(\frac{4b}{b^{2}-1}-\frac{b-1}{b+1}\right)x_{D}+\frac{2b}{b+1}-\frac{b+\left|P-4\tau\right|}{2} & x_{D}<0 \end{cases} \end{equation} \end_inset where \begin_inset Formula $\left|P-4\tau\right|$ \end_inset is the term that causes the \begin_inset Formula $P\mathbin{:}1$ \end_inset tongues to be symmetric across \begin_inset Formula $\tau=\frac{P}{4}$ \end_inset . By setting \begin_inset Formula $x_{D}=0$ \end_inset , we calculate the boundaries of all \begin_inset Formula $P\mathbin{:}1$ \end_inset tongue for odd \begin_inset Formula $P\geq5$ \end_inset and \begin_inset Formula $b\in\left[1,3\right]$ \end_inset as \begin_inset Formula \begin{equation} \mathbb{\tau_{\mathrm{P}}}(b)=\frac{P}{4}\pm\frac{3b-b^{2}}{4\left(b+1\right)} \end{equation} \end_inset We find that this phenomenon persists throughout the system. The vast majority of tongues investigated are bounded by BCSN bifurcations occurring when a \begin_inset Formula $D$ \end_inset crosses \begin_inset Formula $x=0$ \end_inset . For \begin_inset Formula $b>1$ \end_inset , a stable \begin_inset Formula $P\mathbin{:}R$ \end_inset solution and an unstable \begin_inset Formula $P\mathbin{:}R+2$ \end_inset solution undergo a SN bifurcation when a \begin_inset Formula $D$ \end_inset crosses \begin_inset Formula $x=0$ \end_inset if the solution has the symmetry \begin_inset Formula $x(t)=-x(t+\frac{P}{2})$ \end_inset . Otherwise, the stable \begin_inset Formula $P\mathbin{:}R$ \end_inset solution undergoes a SN bifurcation with an unstable \begin_inset Formula $P\mathbin{:}R+1$ \end_inset solution when a \begin_inset Formula $D$ \end_inset crosses \begin_inset Formula $x=0$ \end_inset . \end_layout \begin_layout Standard For \begin_inset Formula $b<1$ \end_inset , the mechanism by which BCSN bifurcations occur is slightly different. Instead of a \begin_inset Formula $D$ \end_inset crossing \begin_inset Formula $x=0$ \end_inset and adding new symbols to the sequence, \begin_inset Formula $D$ \end_inset instead crosses \begin_inset Formula $x=0$ \end_inset by swapping order with an existing \begin_inset Formula $Z$ \end_inset . The stable \begin_inset Formula $5\mathbin{:}1$ \end_inset solution \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D},D,\bar{D}\right]^{-}$ \end_inset undergoes a BCSN bifurcation with the unstable \begin_inset Formula $5\mathbin{:}1$ \end_inset \begin_inset Formula $\left[Z,D,\bar{D},D,\bar{H},\bar{D},D\right]^{-}$ \end_inset at \begin_inset Formula $\left(b,\frac{5-b}{4}\right)$ \end_inset , \begin_inset Formula $b<1$ \end_inset , as shown in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sn_bif" \end_inset (a,c,d). While the unstable \begin_inset Formula $5:1$ \end_inset and \begin_inset Formula $5:3$ \end_inset solutions shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:sn_bif" \end_inset are two different solutions under our sequence representation, they are identical at \begin_inset Formula $b=1$ \end_inset , where a \begin_inset Formula $D$ \end_inset passes through \begin_inset Formula $x=0$ \end_inset without a bifurcation. A noteworthy feature of the BCSN bifurcations at the boundary of the \begin_inset Formula $5\mathbin{:}1$ \end_inset tongue is that the pair of \begin_inset Formula $D$ \end_inset events which collide at \begin_inset Formula $x=0$ \end_inset is different for \begin_inset Formula $b<1$ \end_inset and \begin_inset Formula $b>1$ \end_inset , as seen in Figure Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:sn_bif" \end_inset (a,b,c,e). Generally, for \begin_inset Formula $b<1$ \end_inset , a stable \begin_inset Formula $P\mathbin{:}1$ \end_inset solution undergoes a BCSN bifurcation with an unstable \begin_inset Formula $P\mathbin{:}1$ \end_inset solution. Again, we produce a general form for the curves where a \begin_inset Formula $P\mathbin{:}1$ \end_inset solution undergoes a BCSN bifurcation for odd \begin_inset Formula $P\geq1$ \end_inset and and \begin_inset Formula $b\in\left[0,1\right]$ \end_inset as \begin_inset Formula \begin{equation} \mathbb{\tau_{\mathrm{P}}}(b)=\frac{P\pm b}{4} \end{equation} \end_inset The \begin_inset Formula $1\mathbin{:}1$ \end_inset solution is a special case. The stable \begin_inset Formula $\left[Z,\bar{H},\bar{D}\right]^{-}$ \end_inset solution and the unstable \begin_inset Formula $\left[Z,D,\bar{H}\right]^{-}$ \end_inset undergo a BCSN bifurcation along \begin_inset Formula $\left(b,\frac{n}{2}+\frac{1-b}{4}\right)$ \end_inset , \begin_inset Formula $b<1$ \end_inset , \begin_inset Formula $n\in\mathbb{Z^{\mathrm{+}}}$ \end_inset . The stable \begin_inset Formula $\left[Z,\bar{D},\bar{H}\right]^{-}$ \end_inset solution and the unstable \begin_inset Formula $\left[Z,\bar{H},D\right]^{-}$ \end_inset undergo a BCSN bifurcation along \begin_inset Formula $\left(b,\frac{n}{2}+\frac{1+b}{4}\right)$ \end_inset , \begin_inset Formula $b<1$ \end_inset , \begin_inset Formula $n\in\mathbb{Z^{\mathrm{+}}}$ \end_inset . \end_layout \begin_layout Standard A selection of these curves can be seen plotted in solid white in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset . There is a conspicuous outlier to this pattern. The \begin_inset Formula $3\mathbin{:}1$ \end_inset solution does not undergo a BCSN bifurcation at the right boundary of the \begin_inset Formula $3\mathbin{:}1$ \end_inset tongue for \begin_inset Formula $b>1$ \end_inset . We will now investigate why this is the case. \end_layout \begin_layout Subsection Border-collision torus bifurcation of the \begin_inset Formula $3\mathbin{:}1$ \end_inset solution \end_layout \begin_layout Standard Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset shows that the \begin_inset Formula $3\mathbin{:}1$ \end_inset tongue has the same shape as the other \begin_inset Formula $P\mathbin{:}1$ \end_inset tongues, rooted at \begin_inset Formula $\tau=0.75$ \end_inset . Along some of the boundary, the \begin_inset Formula $3\mathbin{:}1$ \end_inset solution changes to the \begin_inset Formula $3\mathbin{:}3$ \end_inset solution without bifurcation. We test the observed sequences which represent the \begin_inset Formula $3\mathbin{:}3$ \end_inset solution and find that the \begin_inset Formula $3\mathbin{:}3$ \end_inset solution exists in the region \begin_inset Formula $\left[1,3\right]\times\left[0.5,1\right]$ \end_inset outside of the \begin_inset Formula $3\mathbin{:}1$ \end_inset tongue. However, the \begin_inset Formula $3\mathbin{:}3$ \end_inset solution is not always stable; this can be seen in the top half of the \begin_inset Formula $3\mathbin{:}3$ \end_inset region in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset . In order to investigate this phenomenon, we construct a Poincaré map \begin_inset Formula $\mathbb{T}$ \end_inset by a similar method as for the BCSN bifurcation. For \begin_inset Formula $\tau>0.75$ \end_inset , the \begin_inset Formula $3\mathbin{:}1$ \end_inset solution is represented by \begin_inset Formula $\left[Z,\bar{D},D,\bar{H},\bar{D}\right]^{-}$ \end_inset from which we derive \begin_inset Formula $\mathbb{T}^{+}$ \end_inset for \begin_inset Formula $x_{D}\geq0$ \end_inset . For \begin_inset Formula $x_{D}<0$ \end_inset , the sequence changes to \begin_inset Formula $\left[Z,\bar{H},H,\bar{D},\bar{Z},D,Z,\bar{H},\bar{D}\right]^{-}$ \end_inset , from which we derive \begin_inset Formula $\mathbb{T}^{-}$ \end_inset . There is a significant difference however; the \begin_inset Formula $3\mathbin{:}1$ \end_inset solution is one-dimensional, but for \begin_inset Formula $\tau>0.75$ \end_inset , the \begin_inset Formula $3\mathbin{:}3$ \end_inset solution is two-dimensional. This occurs because the additional \begin_inset Formula $\bar{H}$ \end_inset and \begin_inset Formula $H$ \end_inset events occur between the \begin_inset Formula $D$ \end_inset that crosses \begin_inset Formula $x=0$ \end_inset and the \begin_inset Formula $Z$ \end_inset present in both sequences. Therefore, we also need to know the time \begin_inset Formula $t_{\bar{H}}$ \end_inset at which the \begin_inset Formula $\bar{H}$ \end_inset present in both sequences occurs. We derive \begin_inset Formula $\mathbb{T}:\left(x_{D},t_{\bar{H}}\right)^{T}\rightarrow\left(x_{D}^{*},t_{\bar{H}}^{*}-\frac{3}{2}\right)^{T}$ \end_inset as \begin_inset Formula \begin{equation} \mathbb{T}\left(\begin{array}{c} x_{D}\\ t_{\bar{H}} \end{array}\right)=\begin{cases} \left(\begin{array}{cc} -1 & -2\\ \frac{1}{b+1} & \frac{2}{b+1} \end{array}\right)\left(\begin{array}{c} x_{D}\\ t_{\bar{H}} \end{array}\right)+C & x_{D}\geq0\\ \left(\begin{array}{cc} \frac{4b}{b^{2}-1}-1 & -2\\ \frac{1}{b+1} & \frac{2}{b+1} \end{array}\right)\left(\begin{array}{c} x_{D}\\ t_{\bar{H}} \end{array}\right)+C & x_{D}<0 \end{cases} \end{equation} \end_inset where \begin_inset Formula $C=\left(\begin{array}{c} \frac{3-b}{2}\\ \frac{b}{b+1}+\tau-\frac{3}{2} \end{array}\right)$ \end_inset . Note that \begin_inset Formula $\mathbb{T}^{+}$ \end_inset has two rows which are multiples of each other; this is due to the \begin_inset Formula $3\mathbin{:}1$ \end_inset solution being one-dimensional, and results in a \begin_inset Formula $0$ \end_inset eigenvalue. By setting \begin_inset Formula $x_{D}=0$ \end_inset and solving \begin_inset Formula $\mathbb{T}$ \end_inset for \begin_inset Formula $\tau$ \end_inset , we obtain the curve \begin_inset Formula $\tau=\frac{3}{4}+\frac{3b-b^{2}}{4\left(b+1\right)}$ \end_inset , which matches the upper right boundary of the \begin_inset Formula $3\mathbin{:}1$ \end_inset tongue spanning from \begin_inset Formula $\left(1,1\right)$ \end_inset to \begin_inset Formula $\left(3,0.75\right)$ \end_inset . \begin_inset Formula $\mathbb{T}$ \end_inset has a single fixed point which crosses \begin_inset Formula $x_{D}=0$ \end_inset at the boundary of the \begin_inset Formula $3\mathbin{:}1$ \end_inset tongue. For \begin_inset Formula $x_{D}>0$ \end_inset the fixed point is always stable. For \begin_inset Formula $x_{D}<0$ \end_inset the fixed point is stable for \begin_inset Formula $b>2.6038$ \end_inset , and loses stability when a pair of complex conjugate eigenvalues pass through \begin_inset Formula $\left|\lambda\right|=1$ \end_inset , resulting in a NS bifurcation at \begin_inset Formula $b=2.6038$ \end_inset . \begin_inset Formula $\left|\lambda_{\pm}\right|<1$ \end_inset for \begin_inset Formula $b>2.6038$ \end_inset . Therefore, for \begin_inset Formula $\tau>0.75$ \end_inset the \begin_inset Formula $3\mathbin{:}3$ \end_inset solution is stable for \begin_inset Formula $b>2.6038$ \end_inset . This agrees with the shape of the region in which the \begin_inset Formula $3\mathbin{:}3$ \end_inset solution is observed in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../border_collision_bifurcations_in_a_driven_time_delay_system/bct_bifurcation.png width 85col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:bct" \end_inset Border collision Neimark-Sacker (BCNS) bifurcation of the fixed point of the \begin_inset Formula $3\mathbin{:}1$ \end_inset solution. The upper two plots show the BCNS bifurcation of the half-period map \begin_inset Formula $\mathbb{T}$ \end_inset . The lower plot is based on simulation using our iterative map, and shows the stable closed invariant curve expanding from the fixed point from \begin_inset Formula $x=0$ \end_inset . The fixed point is plotted as a blue circle where it is stable, and as a red triangle where it is unstable. \end_layout \end_inset \end_layout \end_inset For \begin_inset Formula $b<2.6038$ \end_inset , we observe an interesting interaction between \begin_inset Formula $\mathbb{T}^{+}$ \end_inset and \begin_inset Formula $\mathbb{T}^{-}$ \end_inset . As \begin_inset Formula $\mathbb{T}^{+}$ \end_inset has a \begin_inset Formula $0$ \end_inset eigenvalue, any point \begin_inset Formula $\left(x_{D},t_{\bar{H}}\right)$ \end_inset for \begin_inset Formula $x_{D}>0$ \end_inset is mapped directly to a nullcline on which the fixed point sits for \begin_inset Formula $x_{D}>0$ \end_inset . The trajectory then undergoes decaying oscillations to the fixed point in the nullcline. However, if the fixed point is close to \begin_inset Formula $x_{D}=0$ \end_inset , \begin_inset Formula $\mathbb{T}^{+}$ \end_inset can map the point into \begin_inset Formula $x_{D}<0$ \end_inset , where the nullcline does not exist. In the absence of a stable fixed point at \begin_inset Formula $x_{D}<0$ \end_inset , the trajectory starts to spiral out to infinity. As this spiral must cross \begin_inset Formula $x_{D}=0$ \end_inset eventually, the trajectory is caught by the nullcline again, causing an interesting half-spiral attraction. When the fixed point is at \begin_inset Formula $x_{D}<0$ \end_inset , the trajectory converges to a stable attractor that strikes the nullcline at multiple points. Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bct" \end_inset \begin_inset Formula $\left(a,b\right)$ \end_inset show the interaction between \begin_inset Formula $\mathbb{T}^{+}$ \end_inset and \begin_inset Formula $\mathbb{T}^{-}$ \end_inset . Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:bct" \end_inset \begin_inset Formula $\left(c\right)$ \end_inset shows the results of simulations in a \begin_inset Formula $\left(b,\tau\right)$ \end_inset sweep that crosses the boundary of the \begin_inset Formula $3\mathbin{:}1$ \end_inset Arnold tongue. The simulated results agree with those derived from \begin_inset Formula $\mathbb{T}$ \end_inset , and confirm that the closed invariant curve is born from the fixed point as it crosses \begin_inset Formula $x_{D}=0$ \end_inset . We consider this to be a border collision Neimark-Sacker (BCNS) bifurcation of \begin_inset Formula $\mathbb{T}$ \end_inset , similar to that seen by \begin_inset CommandInset citation LatexCommand citet key "meiss_neimarksacker_2008" literal "true" \end_inset . It corresponds to a border collision torus (BCT) bifurcation of the \begin_inset Formula $3\mathbin{:}1$ \end_inset solution in the continuous system. The BCT bifurcation is supercritical, as a stable closed invariant curve expands from the fixed point when it becomes unstable. The BCT bifurcation of the \begin_inset Formula $3\mathbin{:}1$ \end_inset solution and the T bifurcation of the \begin_inset Formula $3\mathbin{:}3$ \end_inset solution are plotted in black in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset . \end_layout \begin_layout Subsection Border collision torus bifurcation of the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution \end_layout \begin_layout Standard Now that we have studied the BCT bifurcation of the \begin_inset Formula $3\mathbin{:}1$ \end_inset solution, we return to the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution. Previously we noted that the Poincaré map \begin_inset Formula $\mathbb{P}$ \end_inset only proved the existence of the torus bifurcation along the vertical boundarie s of the locked region. We can now consider the horizontal boundaries in terms of border collisions. Recall that the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution is represented by four different sequences. At \begin_inset Formula $\tau=\frac{n}{2}$ \end_inset , \begin_inset Formula $n\in\mathbb{N}$ \end_inset , the sequence representing the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution changes when a \begin_inset Formula $H$ \end_inset and a \begin_inset Formula $\bar{H}$ \end_inset cross \begin_inset Formula $x=0$ \end_inset . This results in a change in the dimension of \begin_inset Formula $\mathbb{P}$ \end_inset . Let \begin_inset Formula $\mathbb{P}_{n}$ \end_inset denote \begin_inset Formula $\mathbb{P}$ \end_inset when \begin_inset Formula $\mathbb{P}$ \end_inset contains an \begin_inset Formula $n\times n$ \end_inset matrix. When \begin_inset Formula $\tau$ \end_inset increases past \begin_inset Formula $\tau=\frac{n}{2}$ \end_inset , \begin_inset Formula $\mathbb{P}$ \end_inset changes from \begin_inset Formula $\mathbb{P}_{n}$ \end_inset to \begin_inset Formula $\mathbb{P_{\mathrm{n+1}}}$ \end_inset . Let us consider \begin_inset Formula $x_{H}$ \end_inset , the position of the \begin_inset Formula $H$ \end_inset event, as the fixed point of the map. Then \begin_inset Formula $x_{H}=0$ \end_inset when \begin_inset Formula $\mathbb{P}$ \end_inset changes from \begin_inset Formula $\mathbb{P}_{n}$ \end_inset to \begin_inset Formula $\mathbb{P_{\mathrm{n+1}}}$ \end_inset , resulting in a border collision at \begin_inset Formula $x_{H}=0$ \end_inset . Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bct_locked" \end_inset shows four cases that occur in the border collision when we sweep across the border \begin_inset Formula $\tau=1.5$ \end_inset for fixed \begin_inset Formula $b$ \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../border_collision_bifurcations_in_a_driven_time_delay_system/bct_locked_bifurcation.png width 85col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:bct_locked" \end_inset \begin_inset Formula $x$ \end_inset -coordinates of \begin_inset Formula $H$ \end_inset events from simulations. The four plots show the different cases of the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution at the border. \end_layout \end_inset \end_layout \end_inset Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bct_locked" \end_inset (a) shows the stable \begin_inset Formula $1\mathbin{:}1$ \end_inset solution remaining stable, as both \begin_inset Formula $\mathbb{P_{\mathrm{3}}}$ \end_inset and \begin_inset Formula $\mathbb{P_{\mathrm{4}}}$ \end_inset are stable at \begin_inset Formula $b=6.5$ \end_inset . Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bct_locked" \end_inset (b) shows the stable \begin_inset Formula $1\mathbin{:}1$ \end_inset solution becoming unstable, generating a \begin_inset Formula $13\mathbin{:}13$ \end_inset solution in a supercritical BCT bifurcation. This solution exists in one of the vertical stripes we noted in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset . Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bct_locked" \end_inset (c) shows the stable \begin_inset Formula $1\mathbin{:}1$ \end_inset solution becoming unstable, generating an aperiodic solution in a supercritical BCT bifurcation, showing that there are gaps between the vertical stripes. Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bct_locked" \end_inset (d) shows the stable \begin_inset Formula $1\mathbin{:}1$ \end_inset solution becoming unstable; however, the aperiodic solution that the system converges to afterwards is not generated at the border, indicating that the BCT bifurcation is subcritical at this point. Therefore the BCT bifurcation of the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution must change criticality at some point along \begin_inset Formula $\tau=1.5$ \end_inset . The BCT bifurcation of the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution produces the horizontal black lines in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset , completing the boundary of the locked region. \end_layout \begin_layout Standard We examine the difference in maxima of nearby solutions on either side of \begin_inset Formula $\tau=1.5$ \end_inset in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset (b). The point \begin_inset Formula $\left(b,\tau\right)=\left(1.5,3.86\right)$ \end_inset stands out; the difference in maxima is sudden to the left of that point, and gradual to the right. The gradual change in maxima occurs where the BCT bifurcation is supercritical, and the abrupt change in maxima occurs where the BCT bifurcation is subcritical , as illustrated by Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:bct_locked" \end_inset (b-d). We note that \begin_inset Formula $\left(b,\tau\right)=\left(1.5,3.86\right)$ \end_inset forms one corner of a roughly triangular region containing vertical \begin_inset Formula $P\mathbin{:}P$ \end_inset stripes seen in Fig. \begin_inset space ~ \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:period" \end_inset ; every \begin_inset Formula $P\mathbin{:}R$ \end_inset solution in this region is \begin_inset Formula $P\mathbin{:}P$ \end_inset . This leads us to the dashed white curve in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset . We refer to this curve as the \begin_inset Formula $D\bar{D}$ \end_inset curve. To the right of the \begin_inset Formula $D\bar{D}$ \end_inset curve, all \begin_inset Formula $D$ \end_inset events occur at \begin_inset Formula $x<0$ \end_inset and all \begin_inset Formula $\bar{D}$ \end_inset events occur at \begin_inset Formula $x>0$ \end_inset , and so a legal sequence cannot contain a \begin_inset Formula $D$ \end_inset adjacent to a \begin_inset Formula $\bar{D}$ \end_inset ; a \begin_inset Formula $P\mathbin{:}R$ \end_inset solution in this region must, therefore, be \begin_inset Formula $P\mathbin{:}P$ \end_inset . To the left of the \begin_inset Formula $D\bar{D}$ \end_inset curve, a legal sequence representing a solution can contain a \begin_inset Formula $D$ \end_inset adjacent to a \begin_inset Formula $\bar{D}$ \end_inset ; we refer to such a solution as a \begin_inset Formula $D\bar{D}$ \end_inset solution. The point at which the BCT bifurcation changes criticality occurs where the \begin_inset Formula $D\bar{D}$ \end_inset curve intersects the boundary of the locked region. \end_layout \begin_layout Standard As noted previously, when sweeping from right to left, the system converges to the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution until it becomes unstable at the T bifurcation. Now, we see that when sweeping from left to right, the system converges to \begin_inset Formula $D\bar{D}$ \end_inset solutions until they vanish at the \begin_inset Formula $D\bar{D}$ \end_inset curve; we note that a \begin_inset Formula $6\mathbin{:}5$ \end_inset and a \begin_inset Formula $6\mathbin{:}6$ \end_inset solution undergo a BCSN bifurcation at the \begin_inset Formula $D\bar{D}$ \end_inset curve. If the T bifurcation lies to the left of the \begin_inset Formula $D\bar{D}$ \end_inset curve, then the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution remains stable while the \begin_inset Formula $D\bar{D}$ \end_inset solutions exist, resulting in regions of multistability. We observe that this behaviour, together with the change in criticality of the BCT bifurcation, is reminiscent of a Chenciner (generalized Neimark-Sack er) bifurcation, a co-dimension-2 bifurcation that was observed in the smooth system by \begin_inset CommandInset citation LatexCommand citet key "keane_chenciner_2018" literal "true" \end_inset where it produced rich dynamics. \end_layout \begin_layout Standard If the T bifurcation lies to the right of the \begin_inset Formula $D\bar{D}$ \end_inset curve, this produces regions where vertical stripes can occur between the BCT bifurcation of the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution and the \begin_inset Formula $D\bar{D}$ \end_inset curve. Note that the vertical stripes do not stretch all the way between the \begin_inset Formula $D\bar{D}$ \end_inset curve and the BCT bifurcation. There is a second condition that a region must satisfy for the existence of vertical stripes, which is shown by the dashed black curve in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset . Vertical stripes only occur where the order of \begin_inset Formula $D$ \end_inset , \begin_inset Formula $\bar{D}$ \end_inset , \begin_inset Formula $H$ \end_inset , \begin_inset Formula $\bar{H}$ \end_inset symbols in the stable solution is consistent with the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution. The difference between stable \begin_inset Formula $P\mathbin{:}P$ \end_inset solutions and the unstable \begin_inset Formula $1\mathbin{:}1$ \end_inset solution they coexist with lies only in the order of \begin_inset Formula $Z$ \end_inset \begin_inset Formula $H$ \end_inset , \begin_inset Formula $\bar{Z}$ \end_inset and \begin_inset Formula $\bar{H}$ \end_inset symbols. This arises because these solutions are generated when \begin_inset Formula $Z$ \end_inset , \begin_inset Formula $H$ \end_inset , \begin_inset Formula $\bar{Z}$ \end_inset and \begin_inset Formula $\bar{H}$ \end_inset symbols swapped order in the sequence representation of the \begin_inset Formula $1\mathbin{:}1$ \end_inset solution at \begin_inset Formula $\tau=\frac{n}{2}$ \end_inset , \begin_inset Formula $n\in\mathbb{N}$ \end_inset . \end_layout \begin_layout Section Conclusion \end_layout \begin_layout Standard We thoroughly analysed an elementary two-parameter system which combines the effects of time delayed feedback and periodic forcing. In spite of its simplicity, it demonstrates a complex structure of Arnold tongues with zero-width shrinking points and a high degree of multistability. Due to the system being piecewise-linear, we are able to solve the system analytically using an iterative map. We investigate the existence and stability of solutions through the development of a symbolic representation of solutions and the analysis of the subsequently developed Poincaré and border collision maps. This analysis reveals that the Arnold tongues are bounded by curves of border collision saddle-node bifurcations of periodic orbits. Additionally, we find curves of torus bifurcations connected to curves of border collision torus bifurcations, and investigate changes in the criticality of these bifurcations. \end_layout \begin_layout Standard Our analysis sheds new light onto previously obtained results in related smooth systems, particularly the El Niño Southern Oscillation climate model studied by \begin_inset CommandInset citation LatexCommand citet key "keane_delayed_2015" literal "true" \end_inset . Comparing the numerically calculated bifurcation structure found in that paper with the analytically calculated bifurcation structure found here reveals that the prominent features of the smooth system ( \begin_inset CommandInset ref LatexCommand ref reference "eq:1" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "eq:2" \end_inset ) are preserved in the non-smooth limit of \begin_inset Formula $\kappa\rightarrow\infty$ \end_inset . Indeed, the solutions in the smooth system generally appear as \begin_inset Quotes eld \end_inset smoothed out \begin_inset Quotes erd \end_inset counterparts to the piecewise-linear solutions of the non-smooth system. There are some significant differences. The tongues in the smooth system do not feature shrinking points, which are expected only in non-smooth systems, according to \begin_inset CommandInset citation LatexCommand citet key "simpson_resonance_2010" literal "true" \end_inset . Additionally, the smooth system features period doubling bifurcations, which we do not observe in our system. This is likely due to the complete lack of smoothness in our system, which prevents small dynamical variations in feedback strength and forcing strength. However, the analysis presented here does provide new insights into dynamics previously observed numerically \begin_inset CommandInset citation LatexCommand citep key "ghil_delay_2008,keane_delayed_2015" literal "true" \end_inset . \end_layout \begin_layout Standard Both delayed feedback and periodic forcing are very common mathematical model ingredients and can be found in a variety of models used to study, for example, laser dynamics \begin_inset CommandInset citation LatexCommand citep key "sorrentino_effects_2015" literal "true" \end_inset and chimera states \begin_inset CommandInset citation LatexCommand citep key "semenov_delayed_2016" literal "true" \end_inset . The current work reveals the phenomena which are a genuine consequence of this combination. Therefore, we expect this work to be of interest to a wide readership. \end_layout \begin_layout Chapter Dynamics of a band-pass filter system with switched time-delayed feedback \begin_inset CommandInset label LatexCommand label name "chap:Dynamics-of-a" \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash renewcommand{ \backslash leftmark}{Dynamics of a band-pass filter system with switched time-delayed feedback} \end_layout \end_inset \end_layout \begin_layout Standard The material in this chapter is from a paper that I am co-authoring with Lucas Illing and Andreas Amann. My contribution to this paper consisted of deriving the map which analytically models the piecewise-linear system, exploring the parameter space and demonstra ting the dynamics of the underdamped regime, proposing the mechanisms by which observed solutions cease to exist, and numerically determining the criticality of the subcritical torus bifurcations. \end_layout \begin_layout Section Abstract \end_layout \begin_layout Standard We consider a piecewise-linear second-order delay differential equation which is representative of feedback systems with switches (relays) that actuate after a fixed delay. In particular, our model describes a single-input single-output system with delayed self-feedback in which the feedback is a band-pass filtered relay signal. We present a detailed study of periodic solutions and their bifurcations. Starting from an integro-differential model, we show how to reduce the system to a set of finite-dimensional maps. We demonstrate that the stability of solutions can be understood in terms of smooth bifurcations of maps, and the existence of solutions can be understoo d in terms of border-collision bifurcations corresponding to transitions between maps. \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard Many naturally occurring and technological systems evolve based not only on their current state, but also on their state some time in the past, making them time-delay systems. The effect of time delay has been studied in a wide variety of systems, including photonic \begin_inset CommandInset citation LatexCommand cite key "heil_chaos_2001,wieczorek_dynamical_2005" literal "false" \end_inset , optoelectronic \begin_inset CommandInset citation LatexCommand cite key "callan2010broadband,chembo2019optoelectronic,peil2009routes" literal "false" \end_inset , electronic \begin_inset CommandInset citation LatexCommand cite key "larger2013virtual,illing2006ultra" literal "false" \end_inset , neuroscience \begin_inset CommandInset citation LatexCommand cite key "schoell2009time" literal "false" \end_inset , and climate systems \begin_inset CommandInset citation LatexCommand cite key "ryan2020border,keane_investigating_2016,keane_delayed_2015,runge_quantifying_2014" literal "false" \end_inset . Delay differential equations (DDEs) that arise from time-delay systems have, generically, an infinite dimensional state space \begin_inset CommandInset citation LatexCommand cite key "hale2002dynamics" literal "false" \end_inset , which makes analytic study of DDEs challenging, but introduces a wealth of complex dynamic behaviour on multiple time scales \begin_inset CommandInset citation LatexCommand cite key "erneux2009applied,peil2009routes,callan2010broadband,hale2002dynamics,larger2010nonlinear" literal "false" \end_inset . The ubiquity of time delay, and the complexity of the induced dynamics, have made it a phenomenon of great interest to researchers. Time delay plays an important role in applications such as chaos control \begin_inset CommandInset citation LatexCommand cite key "pyragas1992continuous,amann2007some,schoell2008handbook,schoell2016control,hooton2012analytical,illing2007controlling" literal "false" \end_inset , communication \begin_inset CommandInset citation LatexCommand cite key "roy1999chaotic" literal "false" \end_inset as well as computing \begin_inset CommandInset citation LatexCommand cite key "roy2019delayed" literal "false" \end_inset . \end_layout \begin_layout Standard The combination of time-delayed feedback and nonlinearity has been observed to give rise to a wealth of possible dynamics, such as multiple coexisting attractors and oscillatory behavior that ranges from periodic waveforms to high-dimensional chaos as well as hybrid states such as chaotic breathers \begin_inset CommandInset citation LatexCommand cite key "erneux2009applied,peil2009routes,callan2010broadband,larger2010nonlinear,hale2002dynamics" literal "false" \end_inset . In many applications, nonlinear terms containing time delays are well approxima ted by functions that take on discrete values. An important example are relay control systems \begin_inset CommandInset citation LatexCommand cite key "sieber2010dynamics,sieber2006dynamics,barton2006periodic" literal "false" \end_inset . In this approximation, DDEs become non-smooth. Often this leads to significant simplifications of the analytic treatment with the trade-off being that the interplay between the discontinuous nonlinear ities (switching events) and delay in the switching functions leads to new types of bifurcation scenarios, referred to as discontinuity-induced bifurcatio ns or border-collision bifurcations \begin_inset CommandInset citation LatexCommand cite key "barton2005explicit,banerjee_border_1999,di_bernardo_bifurcations_2008,colombo_bifurcations_2012,barton2006periodic" literal "false" \end_inset . As the study of such bifurcation scenarios is still relatively recent, there is still much to be learned by studying the dynamics that can arise from simple non-smooth delay differential equations. \end_layout \begin_layout Standard Let us consider a simple second-order delay differential equation derived from a widespread technology in communications and signal processing. A \emph on band-pass filter \emph default is a system or device that takes a input signal, passes frequencies within a bandwidth centred on a centre frequency, and attenuates (diminishes) frequencies outside that range. Such devices are used broadly in signal processing. The dynamics of a band-pass filter can be described by the system of equations \begin_inset CommandInset citation LatexCommand cite key "udaltsov2002bandpass,illing2005hopf,callan2010broadband" literal "false" \end_inset \begin_inset Formula \begin{equation} \begin{aligned}Q\Omega^{-1}\dot{x} & =-x-y+f\left(t\right)\\ \dot{y} & =Q\Omega x \end{aligned} \label{eq:dde} \end{equation} \end_inset where \begin_inset Formula $f\left(t\right)$ \end_inset is the input signal, \begin_inset Formula $\Omega$ \end_inset is the centre frequency, \begin_inset Formula $Q$ \end_inset is the quality factor, defined as the ratio of the centre frequency divided by the bandwidth. A complete derivation of this system can be found in Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Derivation-of-the" plural "false" caps "false" noprefix "false" \end_inset . When we drive the band-pass filter Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dde" plural "false" caps "false" noprefix "false" \end_inset ) with input signal \begin_inset Formula $f$ \end_inset \begin_inset Formula $\left(t\right)=\cos\left(\omega t\right)$ \end_inset , the output \begin_inset Formula $x\left(t\right)$ \end_inset is a signal with frequency \begin_inset Formula $\omega$ \end_inset whose amplitude depends on \begin_inset Formula $Q$ \end_inset , and \begin_inset Formula $\omega$ \end_inset relative to \begin_inset Formula $\Omega$ \end_inset , as shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Band-pass-filter-output" plural "false" caps "false" noprefix "false" \end_inset . Outside of a given bandwidth determined by \begin_inset Formula $Q$ \end_inset , the amplitude of the output signal decreases according to a power law as \begin_inset Formula $\omega$ \end_inset moves away from \begin_inset Formula $\Omega$ \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../lucas_system/fig_bandpass_filter.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Band-pass filter output amplitude with input \begin_inset Formula $f$ \end_inset \begin_inset Formula $\left(t\right)=\cos\left(\omega t\right)$ \end_inset for varying input frequency \begin_inset Formula $\omega$ \end_inset , where the centre frequency \begin_inset Formula $\Omega=1$ \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:Band-pass-filter-output" \end_inset \end_layout \end_inset \end_layout \end_inset To explore the effects of delayed relay feedback in the context of a band-pass filter, we drive Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dde" plural "false" caps "false" noprefix "false" \end_inset ) with time-delayed switched negative feedback \begin_inset Formula \begin{equation} f\left(t\right)=-\text{sgn}\left(x\left(t-\tau\right)\right)\label{eq:feedback} \end{equation} \end_inset where \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \text{sgn}\left(x\right)=\begin{cases} +1 & x>0\\ -1 & x<0\\ 0 & x=0 \end{cases}\label{eq:signum} \end{equation} \end_inset yielding the non-smooth delay-differential equation \begin_inset Formula \begin{equation} \begin{aligned}Q\Omega^{-1}\dot{x} & =-x-y-\text{sgn}\left(x\left(t-\tau\right)\right)\\ \dot{y} & =Q\Omega x \end{aligned} \label{eq:dde-2} \end{equation} \end_inset As a consequence of the discontinuous input signal, Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dde-2" plural "false" caps "false" noprefix "false" \end_inset ) can be simplified by rescaling \begin_inset Formula $t$ \end_inset w.r.t. \begin_inset Formula $\tau$ \end_inset , yielding the system that we study in this paper \begin_inset Formula \begin{equation} \begin{aligned}Q\Omega^{-1}\dot{x} & =-x-y-\text{sgn}\left(x\left(t-1\right)\right)\\ \dot{y} & =Q\Omega x \end{aligned} \label{eq:dde_master} \end{equation} \end_inset The system is equivariant, possessing the symmetry \begin_inset Formula \begin{equation} \left(\begin{array}{c} x\left(t\right)\\ y\left(t\right) \end{array}\right)\rightarrow\left(\begin{array}{c} -x\left(t\right)\\ -y\left(t\right) \end{array}\right) \end{equation} \end_inset which is a prerequisite for some of the dynamical behaviour observed in this system. While \begin_inset Formula $\text{sgn}\left(x\left(t-1\right)\right)$ \end_inset is constant, the feedback is constant and the system is linear. However, the system is discontinuous at changes in the feedback, and so the system is piecewise-linear. Therefore, while delay differential equations are generically infinite-dimensio nal, the state of this system is fully specified by knowing \begin_inset Formula $x\left(t\right)$ \end_inset , \begin_inset Formula $y\left(t\right)$ \end_inset , and the \emph on zero elements \emph default , which we define in the same manner as we did in Chapter \begin_inset CommandInset ref LatexCommand ref reference "chap:Border-collision-bifurcations-in" plural "false" caps "false" noprefix "false" \end_inset as the times at which the trajectory generated by Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dde_master" plural "false" caps "false" noprefix "false" \end_inset ) passed through \begin_inset Formula $x=0$ \end_inset in the time interval \begin_inset Formula $\left(t-1,t\right]$ \end_inset . Once again, we have no \emph on a priori \emph default assumptions of how many zero elements there can be at any given time \begin_inset Formula $t$ \end_inset . Therefore, we let \begin_inset Formula $n$ \end_inset be the number of zero elements, and define the state of the system \begin_inset Formula $S\left(t\right)$ \end_inset as \begin_inset Formula \begin{equation} S(t)=\left(x,y;z_{0},z_{1},...,z_{n-1}\right)\label{eq:state} \end{equation} \end_inset where \begin_inset Formula $\left(x,y\right)\in\mathbb{R}^{2}$ \end_inset is the position at time \begin_inset Formula $t$ \end_inset and \begin_inset Formula $t-\tau0$ \end_inset , \begin_inset Formula $Re\left(\lambda_{\pm}\right)<0$ \end_inset , so that the fixed point is always stable. In the overdamped regime \begin_inset Formula $Q<\frac{1}{2}$ \end_inset , the eigenvalues are real and the fixed point is a sink. In the underdamped regime \begin_inset Formula $Q>\frac{1}{2}$ \end_inset , the eigenvalues are complex and the fixed point is a stable spiral; this causes the trajectory to wind around the fixed point as it approaches. Thus, the fixed point is always attracting, and any more complicated dynamics must arise from a combination of switching and the transient behaviour of the linear system. In our analysis, we will use the fixed point of the linear subsystem as a point of reference; it is not a fixed point of the full system. \end_layout \begin_layout Subsection Reducing the system to a map \end_layout \begin_layout Standard We can make use of the piecewise-linear nature of the system to simulate it analytically. A linear system \begin_inset Formula \begin{equation} \boldsymbol{\dot{x}}=A\boldsymbol{x} \end{equation} \end_inset with fixed point at the origin has solution \begin_inset Formula \begin{equation} \boldsymbol{x}\left(t+T\right)=F_{T}\left(\boldsymbol{x}\left(t\right)\right)\label{eq:linear_system} \end{equation} \end_inset for some \begin_inset Formula $T>0$ \end_inset where \begin_inset Formula $F_{\tau}$ \end_inset is defined using the matrix exponential \begin_inset CommandInset citation LatexCommand cite key "moler2003nineteen" literal "false" \end_inset \begin_inset Formula \begin{equation} F_{T}\left(\boldsymbol{x}\right)=e^{AT}\boldsymbol{x}\label{eq:F} \end{equation} \end_inset which is calculated in Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Calculating-the-matrix" plural "false" caps "false" noprefix "false" \end_inset . We introduce a change of variables \begin_inset Formula $G$ \end_inset that shifts the fixed point to the origin \begin_inset Formula \begin{equation} G\left(\begin{array}{c} x\\ y \end{array}\right)=\left(\begin{array}{c} x\\ y+\text{sgn}\left(x\left(t-1\right)\right) \end{array}\right)\label{eq:change_of_variables} \end{equation} \end_inset \begin_inset Formula $G$ \end_inset remains constant as long as the feedback remains constant. Thus, we can simulate the system analytically between changes in the feedback by \begin_inset Formula \begin{equation} \left(\begin{array}{c} x\left(t+T\right)\\ y\left(t+T\right) \end{array}\right)=G^{-1}F_{T}G\left(\begin{array}{c} x\left(t\right)\\ y\left(t\right) \end{array}\right)\label{eq:analytic map} \end{equation} \end_inset where \begin_inset Formula $T=\tau+z_{0}$ \end_inset . This allows us simulate the system exactly as a map between changes in the feedback. However, to know when the changes in feedback occur, we also need to know when the trajectory generated by the system has passed through \begin_inset Formula $x=0$ \end_inset . This shows that the key events in this system are zero crossings and feedback changes \begin_inset CommandInset citation LatexCommand cite key "ryan2020border" literal "false" \end_inset . Thus, we simulate the system as a map between \begin_inset Formula $Z$ \end_inset events, or zero crossings, where the trajectory passes through \begin_inset Formula $x-0$ \end_inset and a new history element is appended to \begin_inset Formula $S$ \end_inset , and \begin_inset Formula $H$ \end_inset events, where the oldest history element is removed from \begin_inset Formula $S$ \end_inset and the feedback changes. The \begin_inset Formula $H$ \end_inset events, in particular, are of great interest when we come to analyse the dynamics of the system. \end_layout \begin_layout Subsection Sample solutions \end_layout \begin_layout Standard We now present sample solutions which demonstrate some of the characteristic features of this system. Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Sample-solutions" plural "false" caps "false" noprefix "false" \end_inset shows eight stable solutions found by simulating the system from different initial conditions and parameters \begin_inset Formula $\left(\Omega,Q\right)$ \end_inset . In each figure, \begin_inset Formula $t$ \end_inset has been shifted so that a change in feedback occurs at \begin_inset Formula $t=0$ \end_inset and thus, a zero crossing occurs at \begin_inset Formula $t=-1$ \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../lucas_system/fig_sample_solutions.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Sample solutions, excluding transients, obtained from simulating the system from different initial conditions. In \begin_inset Formula $\left(a\right)$ \end_inset - \begin_inset Formula $\left(c\right)$ \end_inset , \begin_inset Formula $\left(Q,\Omega\right)=\left(0.45,3\pi\right)$ \end_inset . In \begin_inset Formula $\left(d\right)$ \end_inset - \begin_inset Formula $\left(h\right)$ \end_inset , \begin_inset Formula $Q=1.5$ \end_inset , and \begin_inset Formula $\Omega$ \end_inset increases from \begin_inset Formula $\Omega=0.65\pi$ \end_inset in \begin_inset Formula $\left(d\right)$ \end_inset to \begin_inset Formula $\Omega=1.05\pi$ \end_inset in \begin_inset Formula $\left(e\right)$ \end_inset , \begin_inset Formula $\Omega=1.65\pi$ \end_inset in \begin_inset Formula $\left(f\right)$ \end_inset , \begin_inset Formula $\Omega=2.1\pi$ \end_inset in \begin_inset Formula $\left(g\right)$ \end_inset and \begin_inset Formula $\Omega=4.7138\pi$ \end_inset in \begin_inset Formula $\left(h\right)$ \end_inset . In the \begin_inset Formula $\left(x,y\right)$ \end_inset plots, the fixed point of the linear subsystem is plotted in red. \begin_inset CommandInset label LatexCommand label name "fig:Sample-solutions" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard Solutions \begin_inset Formula $\left(a\right)$ \end_inset - \begin_inset Formula $\left(g\right)$ \end_inset are periodic solutions; a periodic solution with period \begin_inset Formula $P$ \end_inset is a trajectory that follows a cycle such that \begin_inset Formula $S(t+P)=S(t)$ \end_inset . Solution \begin_inset Formula $\left(a\right)$ \end_inset demonstrates behaviour that is characteristic of the overdamped regime. The trajectory decays exponentially towards the fixed point until the feedback changes and the fixed point flips, then the trajectory decays towards the fixed point again, passing through \begin_inset Formula $x=0$ \end_inset shortly after the feedback changes. This solution is multistable with higher-frequency solutions \begin_inset Formula $\left(b\right)$ \end_inset and \begin_inset Formula $\left(c\right)$ \end_inset . \end_layout \begin_layout Standard In order to better capture the difference between multistable solutions, we introduce the \emph on mode \emph default \begin_inset Formula $\nu$ \end_inset . \begin_inset Formula $\nu$ \end_inset can be defined as the number of times the solution passes through \begin_inset Formula $x=0$ \end_inset in an open time interval of length \begin_inset Formula $1$ \end_inset that begins with a zero crossing; in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Sample-solutions" plural "false" caps "false" noprefix "false" \end_inset , the interval \begin_inset Formula $\left(-1,0\right)$ \end_inset is convenient for this. Solutions \begin_inset Formula $\left(a\right)$ \end_inset - \begin_inset Formula $\left(c\right)$ \end_inset have mode \begin_inset Formula $\nu=0$ \end_inset , \begin_inset Formula $\nu=2$ \end_inset and \begin_inset Formula $\nu=4$ \end_inset respectively. This definition is equivalent to defining \begin_inset Formula $\nu$ \end_inset as the number of zero elements ( \begin_inset Formula $n$ \end_inset ) in \begin_inset Formula $S$ \end_inset \begin_inset Formula $\left(t\right)$ \end_inset at a time \begin_inset Formula $t$ \end_inset when the feedback changes. Defining \begin_inset Formula $\nu$ \end_inset in this way allows us to compare these multistable solutions to the \begin_inset Formula $1:1$ \end_inset solution analysed in the previous chapter. Instead of having a solution whose appearance changed periodically with \begin_inset Formula $\tau$ \end_inset and whose stability changed according to a Poincaré map with \begin_inset Formula $n=\left\lfloor 2\tau\right\rfloor $ \end_inset variables, here we have multistable solutions of varying \begin_inset Formula $\nu$ \end_inset whose stability depends on a Poincaré map with \begin_inset Formula $\nu+2$ \end_inset variables. \end_layout \begin_layout Standard The underdamped regime features significant multistability, but also contains interesting dynamics not present in the overdamped regime. Solutions \begin_inset Formula $\left(d\right)$ \end_inset - \begin_inset Formula $\left(f\right)$ \end_inset show what happens to the \begin_inset Formula $\nu=0$ \end_inset solution as \begin_inset Formula $\Omega$ \end_inset is varied with \begin_inset Formula $Q$ \end_inset fixed. Solution \begin_inset Formula $\left(d\right)$ \end_inset demonstrates behaviour that is characteristic of the underdamped regime; the trajectory decays towards the fixed point while winding around it until the feedback changes and the fixed point flips, and the trajectory winds around the fixed point again, passing through \begin_inset Formula $x=0$ \end_inset along the way. As \begin_inset Formula $\Omega$ \end_inset increases, the frequency of the solution increases, and the trajectory winds further around the fixed point before the feedback changes; in \begin_inset Formula $\left(e\right)$ \end_inset , the feedback changes just before the trajectory passes through \begin_inset Formula $x=0$ \end_inset . In \begin_inset Formula $\left(f\right)$ \end_inset , \begin_inset Formula $\Omega$ \end_inset has increased further, and the trajectory passes through \begin_inset Formula $x=0$ \end_inset \emph on before \emph default the feedback changes. This continous deformation of the solution as the parameters are varied causes \begin_inset Formula $\nu$ \end_inset to increase from \begin_inset Formula $0$ \end_inset to \begin_inset Formula $1$ \end_inset . This pattern of behaviour is consistent for all solutions in the underdamped regime; for \begin_inset Formula $m\in\mathbb{Z^{+}}$ \end_inset , a \begin_inset Formula $\nu=2m$ \end_inset solution continuously changes to \begin_inset Formula $\nu=2m+1$ \end_inset . This has a significant impact on the solution; when the change in feedback passes through \begin_inset Formula $x=0$ \end_inset , the dimension of the solution increases by 1. Due to the importance of feedback changes, we will make constant reference to the point at which it changes. We define \begin_inset Formula $\left(x_{H},y_{H}\right)^{T}$ \end_inset as the position at which the feedback changes and a history element is deleted from the state \begin_inset Formula $S$ \end_inset . We can calculate where in the \begin_inset Formula $\left(\Omega,Q\right)$ \end_inset plane \begin_inset Formula $x_{H}=0$ \end_inset for a \begin_inset Formula $\nu=2m$ \end_inset solution by noting that this occurs when the elapsed time between zero crossings is \begin_inset Formula $\frac{1}{2m+1}$ \end_inset , as is visible in solution \begin_inset Formula $\left(e\right)$ \end_inset . Therefore, we can calculate curves in the \begin_inset Formula $\left(\Omega,Q\right)$ \end_inset plane along which solutions transition smoothly from \begin_inset Formula $\nu=2m$ \end_inset to \begin_inset Formula $\nu=2m+1$ \end_inset by equating \begin_inset Formula \begin{equation} \frac{1}{2m+1}=\frac{\pi}{\frac{\Omega}{2Q}\sqrt{4Q^{2}-1}} \end{equation} \end_inset which yields \begin_inset Formula \begin{align} \Omega & =\frac{2Q\pi\left(2m+1\right)}{\sqrt{4Q^{2}-1}}\label{eq:smooth_nu_transition} \end{align} \end_inset \end_layout \begin_layout Standard Solutions \begin_inset Formula $\left(g\right)$ \end_inset and \begin_inset Formula $\left(h\right)$ \end_inset show more unusual types of stable solution. Solution \begin_inset Formula $\left(g\right)$ \end_inset shows what happens as we continue to increase \begin_inset Formula $\Omega$ \end_inset from solution \begin_inset Formula $\left(f\right)$ \end_inset ; the symmetry of the solution is broken, yielding an asymmetric solution. This asymmetric solution is observed for a small window in \begin_inset Formula $\Omega$ \end_inset before the trajectory falls off this solution and changes to a \begin_inset Formula $\nu=2$ \end_inset solution. However, that same \begin_inset Formula $\nu=2$ \end_inset solution, upon being swept up to larger \begin_inset Formula $\Omega,$ \end_inset does not break symmetry. Instead, it first transitions smoothly to a \begin_inset Formula $\nu=3$ \end_inset solution as the frequency increases, before changing to an aperiodic solution \begin_inset Formula $\left(h\right)$ \end_inset , which is likewise observed briefly before the trajectory falls off this solution and changes to a \begin_inset Formula $\nu=4$ \end_inset solution. \end_layout \begin_layout Subsection Multistability \end_layout \begin_layout Standard We now examine the multistability observed in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Sample-solutions" plural "false" caps "false" noprefix "false" \end_inset in greater detail. Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_sweep" plural "false" caps "false" noprefix "false" \end_inset show the results of sweeping from \begin_inset Formula $\Omega=12\pi$ \end_inset down to \begin_inset Formula $\Omega=0$ \end_inset and back up for fixed \begin_inset Formula $Q$ \end_inset , beginning at \begin_inset Formula $\Omega=12\pi$ \end_inset with the stable \begin_inset Formula $\nu=12$ \end_inset solution. \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../lucas_system/fig_parameter_space_sweep.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Numerical sweeps of the \begin_inset Formula $\left(\Omega,Q\right)$ \end_inset plane, where the colour indicates the frequency of the swept solution. The white arrow indicates the sweep direction. The dashed white lines show the \begin_inset Formula $Q$ \end_inset values along which we conduct deeper analysis in the overdamped ( \begin_inset Formula $Q=0.45$ \end_inset ) and underdamped ( \begin_inset Formula $Q=1.5$ \end_inset ) regimes. \begin_inset CommandInset label LatexCommand label name "fig:parameter_sweep" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard Consider Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_sweep" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(a\right)$ \end_inset . As we sweep down in \begin_inset Formula $\Omega$ \end_inset , the frequency of the swept solution decreases in large jumps, between which the decrease in frequency is smooth and relatively small. These large jumps in frequency coincide with jumps in the observed value of \begin_inset Formula $\nu$ \end_inset . Note the inconsistent horizontal lines in the lower third of the plot; these artefacts occur where the swept solution drops from an unstable high \begin_inset Formula $\nu$ \end_inset solution to a stable low \begin_inset Formula $\nu$ \end_inset solution, skipping over an stable intermediate \begin_inset Formula $\nu$ \end_inset solution. For example, the largest value of \begin_inset Formula $Q$ \end_inset at which this is observed is around \begin_inset Formula $Q\simeq0.6,$ \end_inset where the swept solution goes from \begin_inset Formula $\nu=12$ \end_inset to \begin_inset Formula $\nu=8$ \end_inset , skipping \begin_inset Formula $\nu=10$ \end_inset and leaving a yellow line surrounded by orange. This may occur because curves of bifurcations grow closer together for small \begin_inset Formula $Q$ \end_inset , leaving the basin of attraction of the intermediate \begin_inset Formula $\nu$ \end_inset solution relatively small compared to that of the low \begin_inset Formula $\nu$ \end_inset solutions at the point where the high \begin_inset Formula $\nu$ \end_inset solution becomes unstable. \end_layout \begin_layout Standard In Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_sweep" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(b\right)$ \end_inset , we observe a significant difference between the underdamped and overdamped regimes. In the underdamped regime \begin_inset Formula $\left(Q>\frac{1}{2}\right)$ \end_inset , as we sweep up in \begin_inset Formula $\Omega$ \end_inset , the frequency of the swept solution increases in large jumps, between which the increase in frequency is smooth and relatively small. These large jumps in frequency coincide with jumps in the observed value of \begin_inset Formula $\nu$ \end_inset . If we combine this observation with our observations from the Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_sweep" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(a\right)$ \end_inset , we can deduce that solutions of given \begin_inset Formula $\nu$ \end_inset (and hence are within a given frequency band) are stable within some band centred on some value of \begin_inset Formula $\Omega$ \end_inset , and that this band is wider for small \begin_inset Formula $Q$ \end_inset . This makes some kind of intuitive sense; bandpass filters attenuate signals outside a given frequency band, so in our bandpass filter with self-feedback, we observe solutions outside a given frequency band becoming unstable. However, this does not happen in the overdamped regime! We observe that the \begin_inset Formula $\nu=0$ \end_inset solution shown does not become unstable as \begin_inset Formula $\Omega$ \end_inset increases; as \begin_inset Formula $\Omega$ \end_inset increases in the overdamped regime, the multistability increases as more and more solutions become stable. \end_layout \begin_layout Standard Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_sweep" plural "false" caps "false" noprefix "false" \end_inset demonstrates that the system exhibits extensive multistability and a variety of types of solution. However, we have not yet determined \emph on why \emph default the swept solution ceases to follow a particular solution. To find out whether this is due to a loss of stability or a loss of existence, we make use of our iterative map of the system to numerically continue a stable solution into regions where they are unstable. In Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:frequency_stability_existence" plural "false" caps "false" noprefix "false" \end_inset , we plot the frequency of the periodic solutions observed through sweeping, where solid lines indicate stable solutions and dashed lines indicate unstable solutions. \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../lucas_system/fig_frequency_stability_existence.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Frequency of solutions plotted against \begin_inset Formula $\Omega$ \end_inset for different \begin_inset Formula $\nu$ \end_inset solutions. Solid lines show where a solution is stable, dashed lines show where the solution is unstable. Triangles show where \begin_inset Formula $\nu$ \end_inset changes from even to odd, circles show where the solution undergoes a torus bifurcation, and squares show where a solution vanishes in a border-collision saddle-node bifurcation of periodic orbits. \begin_inset CommandInset label LatexCommand label name "fig:frequency_stability_existence" \end_inset \end_layout \end_inset \end_layout \end_inset We observe that in the overdamped regime shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:frequency_stability_existence" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(b\right)$ \end_inset , as we sweep down in \begin_inset Formula $\Omega$ \end_inset , the swept solution decreases in frequency when higher frequency solutions become unstable. The \begin_inset Formula $\nu=0$ \end_inset solution is not unique in remaining stable as \begin_inset Formula $\Omega$ \end_inset increases, though it is unique in never becoming unstable as \begin_inset Formula $\Omega$ \end_inset decreases. We also note that these solutions continue to exist as \begin_inset Formula $\Omega\rightarrow\infty$ \end_inset , and for a solution with \begin_inset Formula $\nu=2m$ \end_inset , the frequency tends to \begin_inset Formula $\frac{2m+1}{2}$ \end_inset as \begin_inset Formula $\Omega\rightarrow\infty$ \end_inset . \end_layout \begin_layout Standard Once again, the underdamped regime, shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:frequency_stability_existence" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(a\right)$ \end_inset is more complicated. Like the overdamped regime, as we sweep down in \begin_inset Formula $\Omega$ \end_inset , the swept solution jumps down in frequency when higher frequency solutions become unstable. However, as we sweep upwards, the swept solution jumps up in frequency as lower frequency solutions become unstable. Furthermore, as \begin_inset Formula $\Omega$ \end_inset increases further, the unstable solutions cease to exist. \end_layout \begin_layout Standard We can show a more complete picture of the dynamics in the \begin_inset Formula $\left(\Omega,Q\right)$ \end_inset plane by focusing on solutions with specific values of \begin_inset Formula $\nu$ \end_inset . Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_space_solution_chart" plural "false" caps "false" noprefix "false" \end_inset shows the existence and stability for the \begin_inset Formula $\nu=0,1$ \end_inset solution and the \begin_inset Formula $\nu=2,3$ \end_inset solution. \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../lucas_system/fig_parameter_space_solution_chart.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Existence and stability in the \begin_inset Formula $\left(\Omega,Q\right)$ \end_inset plane for \begin_inset Formula $\left(a\right)$ \end_inset the \begin_inset Formula $\nu=0,1$ \end_inset solution and \begin_inset Formula $\left(b\right)$ \end_inset the \begin_inset Formula $\nu=2,3$ \end_inset solution. The white circles indicate the location of the torus bifurcations shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Subcritical-Hopf-bifurcation" plural "false" caps "false" noprefix "false" \end_inset and Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Supercritical-Hopf-bifurcation" plural "false" caps "false" noprefix "false" \end_inset . The pitchfork bifurcation shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:pitchfork" plural "false" caps "false" noprefix "false" \end_inset takes place near the top of the black triangular region. The black square indicates the location of the saddle-node bifurcation shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Border-collision-saddle-node-bif" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:parameter_space_solution_chart" \end_inset \end_layout \end_inset \end_layout \end_inset We find that there is a region in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_space_solution_chart" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(a\right)$ \end_inset where the symmetric \begin_inset Formula $\nu=1$ \end_inset solution becomes asymmetric, as seen in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Sample-solutions" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(g\right)$ \end_inset . There should also be an extremely narrow band in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_space_solution_chart" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(b\right)$ \end_inset where the \begin_inset Formula $\nu=3$ \end_inset solution becomes unstable and a stable closed invariant curve exists, as seen in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Sample-solutions" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(h\right)$ \end_inset ; however, it is too narrow to plot clearly at this scale. To understand the cause of these changes in stability and existence, we consider the dynamics of this system. \end_layout \begin_layout Section Smooth dynamics \end_layout \begin_layout Subsection Subcritical torus bifurcations of even \begin_inset Formula $\nu\geq2$ \end_inset solutions \end_layout \begin_layout Standard As \begin_inset Formula $\Omega$ \end_inset decreases, all even \begin_inset Formula $\nu\geq2$ \end_inset solutions become unstable along curves in the \begin_inset Formula $\left(\Omega,Q\right)$ \end_inset plane that appear to branch out from the the origin, without generating any other observable phenomena. To understand why this occurs, we experimentally map out the basin of attractio n of the stable \begin_inset Formula $\nu=2$ \end_inset solution as we approach the curve along which it becomes unstable. We do this by simulating the \begin_inset Formula $\nu=2$ \end_inset solution, perturbing the parameters just across the curve, allowing the system to evolve for a period of time, then resetting the parameters and observing whether the system restabilized to the \begin_inset Formula $\nu=2$ \end_inset solution. Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Subcritical-Hopf-bifurcation" plural "false" caps "false" noprefix "false" \end_inset shows the results of this experiment as a Poincaré section through \begin_inset Formula $x_{H}$ \end_inset \begin_inset CommandInset citation LatexCommand cite key "strogatz2018nonlinear" literal "false" \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../lucas_system/fig_subcritical_hopf_overdamped.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout The subcritical torus bifurcation of the \begin_inset Formula $\nu=2$ \end_inset solution in the overdamped regime indicated by a white circle in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_space_solution_chart" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:Subcritical-Hopf-bifurcation" \end_inset \end_layout \end_inset \end_layout \end_inset The basin of attraction takes the characteristic shape of a subcritical Neimark-Sacker or secondary Hopf bifurcation; as we approach the bifurcation curve, an unstable closed invariant curve (inferred from our experiment and plotted in dashed red) collapses onto the stable solution, causing it to become unstable. Thus, as we sweep down in \begin_inset Formula $\Omega$ \end_inset , the periodic solutions become unstable through subcritical torus bifurcations. \end_layout \begin_layout Subsection Supercritical torus bifucations of odd \begin_inset Formula $\nu\geq3$ \end_inset solutions \end_layout \begin_layout Standard As \begin_inset Formula $\Omega$ \end_inset increases, all odd \begin_inset Formula $\nu$ \end_inset solutions eventually become unstable. For \begin_inset Formula $\nu\geq3$ \end_inset , when they become unstable, the Poincaré section through \begin_inset Formula $x_{H}$ \end_inset generates a stable closed invariant curve, as shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Supercritical-Hopf-bifurcation" plural "false" caps "false" noprefix "false" \end_inset , where we follow the \begin_inset Formula $\nu=3$ \end_inset solution which corresponds to the \begin_inset Formula $\nu=2$ \end_inset solution shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Subcritical-Hopf-bifurcation" plural "false" caps "false" noprefix "false" \end_inset . This is consistent with a supercritical Neimark-Sacker bifurcation, and so the periodic solution undergoes a supercritical torus bifurcation, generatin g the aperiodic solution observed in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Sample-solutions" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(h\right)$ \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../lucas_system/fig_supercritical_hopf.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout The supercritical torus bifurcation of the \begin_inset Formula $\nu=3$ \end_inset solution indicated by a white circle in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_space_solution_chart" plural "false" caps "false" noprefix "false" \end_inset . In (b), the torus is plotted in the \begin_inset Formula $\left(x_{H},y_{H}\right)$ \end_inset plane for 20 evenly spaced values of \begin_inset Formula $\Omega$ \end_inset between the dotted black lines in (a). \begin_inset CommandInset label LatexCommand label name "fig:Supercritical-Hopf-bifurcation" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Supercritical pitchfork bifurcation of the \begin_inset Formula $\nu=1$ \end_inset solution \end_layout \begin_layout Standard In the case of the \begin_inset Formula $\nu=1$ \end_inset solution, we do not observe a torus bifurcation generating a closed invariant curve. Instead, a supercritical pitchfork bifurcation generates a pair of stable mirrored asymmetric solutions, one of which was shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Sample-solutions" plural "false" caps "false" noprefix "false" \end_inset (g). The bifurcation diagram of the pitchfork bifurcation is shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:pitchfork" plural "false" caps "false" noprefix "false" \end_inset . The speed at which the asymmetric solutions move apart after the bifurcation makes it look superficially like a nonsmooth bifurcation; however, the inset plot in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:pitchfork" plural "false" caps "false" noprefix "false" \end_inset \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none (a), which covers a much smaller range of \begin_inset Formula $\Omega\in\left[6.5278,6.528\right]$ \end_inset shows that they do spread apart smoothly. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename /home/pierce/projects_pierce_ryan/lucas_system/fig_pitchfork_fixed.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Pitchfork bifurcation of the \begin_inset Formula $\nu=1$ \end_inset solution. \begin_inset CommandInset label LatexCommand label name "fig:pitchfork" \end_inset \end_layout \end_inset \end_layout \end_inset An interesting feature visible in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:pitchfork" plural "false" caps "false" noprefix "false" \end_inset \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none (b) \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit is that the asymmetric solutions cease to exist when \begin_inset Formula $x_{H}=0$ \end_inset . This is noteworthy because if \begin_inset Formula $x_{H}$ \end_inset was to pass through \begin_inset Formula $x=0$ \end_inset , it would have significant consequences for the solution; there would be additional zero crossings, and hence, additional feedback changes, which would cause the solution to change in a manner that is continuous at \begin_inset Formula $x_{H}=0$ \end_inset , but not smooth across \begin_inset Formula $x_{H}=0$ \end_inset . In order to consider these effects, we must consider the non-smooth dynamics of the system. \end_layout \begin_layout Section Non-smooth dynamics \end_layout \begin_layout Subsection Border-collision saddle-node bifurcations of symmetric solutions \end_layout \begin_layout Standard In the underdamped regime, all odd \begin_inset Formula $\nu$ \end_inset solutions vanish when \begin_inset Formula $x_{H}=0$ \end_inset . Similar to how we calculated the smooth \begin_inset Formula $\nu$ \end_inset transitions in Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:smooth_nu_transition" plural "false" caps "false" noprefix "false" \end_inset ), we can calculate where this occurs for symmetric solutions by equating \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{1}{2\left(2m+1\right)+1} & =\frac{\pi}{\frac{\Omega}{2Q}\sqrt{4Q^{2}-1}} \end{align} \end_inset which yields \begin_inset Formula \begin{equation} \Omega=\frac{2Q\pi\left(4m+3\right)}{\sqrt{4Q^{2}-1}} \end{equation} \end_inset However, in this case, the solution does not change smoothly when \begin_inset Formula $\nu$ \end_inset changes. Instead, the \begin_inset Formula $x_{H}$ \end_inset that crosses over \begin_inset Formula $x=0$ \end_inset forms the tip of a spike; when this spike reaches \begin_inset Formula $x=0$ \end_inset , it introduces two new zero crossings and hence, two new feedback changes. We use our iterative map to construct this 'crossed-over' \begin_inset Formula $\nu=5$ \end_inset solution, and plot it in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Border-collision-saddle-node-bif" plural "false" caps "false" noprefix "false" \end_inset . The \begin_inset Formula $\nu=1$ \end_inset solution shown here is the same solution that becomes unstable in a pitchfork bifurcation in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:pitchfork" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../lucas_system/fig_border-collision_saddle-node_symmetric.pdf width 85text% \end_inset \begin_inset Caption Standard \begin_layout Plain Layout Degenerate border-collision saddle-node bifurcation of periodic orbits of the \begin_inset Formula $\nu=1$ \end_inset periodic solution indicated by a black square in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:parameter_space_solution_chart" plural "false" caps "false" noprefix "false" \end_inset . The unstable solution observed while stable in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Sample-solutions" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(d\right)-\left(f\right)$ \end_inset is plotted in dashed red. The corresponding crossed-over \begin_inset Formula $\nu=5$ \end_inset solution is plotted in dash-dot black. \begin_inset CommandInset label LatexCommand label name "fig:Border-collision-saddle-node-bif" \end_inset \end_layout \end_inset \end_layout \end_inset A close up of the spike crossing over is shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Border-collision-saddle-node-bif" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(c\right)$ \end_inset (recall that the solution winds anticlockwise in the \begin_inset Formula $\left(x,y\right)$ \end_inset plane). Initially, the trajectory moves slowly as it decays close to the fixed point, before the feedback changes and the trajectory moves rapidly to decay towards the flipped fixed point. The corresponding change in feedback occurs close to \begin_inset Formula $x=0$ \end_inset for \begin_inset Formula $y<0$ \end_inset . We find that as \begin_inset Formula $\Omega$ \end_inset increases, the crossed-over solution and the original solution collide and vanish in a degenerate border-collision saddle-node (BCSN) bifurcation of periodic orbits between two unstable solutions, as shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Border-collision-saddle-node-bif" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(a\right)$ \end_inset . \end_layout \begin_layout Standard As noted previously, Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:pitchfork" plural "false" caps "false" noprefix "false" \end_inset suggests that non-smooth dynamics are responsible for the vanishing \begin_inset Formula $\nu=1$ \end_inset asymmetric solution. It would be reasonable to expect that a similar BCSN bifurcation of periodic orbits is the cause, and proving this would be of further interest. \end_layout \begin_layout Subsection Global bifurcation of the symmetric crossed-over solution \end_layout \begin_layout Standard The discovery of the crossed-over \begin_inset Formula $\nu=5$ \end_inset solution allows us to explain how the \begin_inset Formula $\nu=1$ \end_inset solution vanishes in a BCSN bifurcation; we now consider how the crossed-over \begin_inset Formula $\nu=5$ \end_inset solution might vanish. In Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Border-collision-saddle-node-bif" plural "false" caps "false" noprefix "false" \end_inset , as \begin_inset Formula $\Omega$ \end_inset decreases, the \begin_inset Formula $x_{H}$ \end_inset that reaches \begin_inset Formula $x=0$ \end_inset at the BCSN bifurcation begins to curve back around towards \begin_inset Formula $x=0$ \end_inset , but the continuation method employed here fails to complete the curve. However, we can note from Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Border-collision-saddle-node-bif" plural "false" caps "false" noprefix "false" \end_inset that as \begin_inset Formula $\Omega$ \end_inset decreases and we move away from the BCSN bifurcation in the parameter space, the \begin_inset Formula $x_{H}$ \end_inset spike on the crossed-over solution moves towards the fixed point as \begin_inset Formula $x_{H}\rightarrow0$ \end_inset . While the spike is shorter, it occurs closer to the fixed point, and so the corresponding feedback change gets longer, as seen in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Border-collision-saddle-node-bif" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(b\right)$ \end_inset . We can consider the limiting behaviour of this solution where the tip of the \begin_inset Formula $x_{H}$ \end_inset spike reaches the fixed point, which we plot in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:limiting-behaviour" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../lucas_system/fig_homoclinic_bifurcation?.pdf width 42.5text% \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Caption Standard \begin_layout Plain Layout The limiting behaviour of the crossed-over \begin_inset Formula $\nu=5$ \end_inset counterpart to the observed \begin_inset Formula $\nu=1$ \end_inset symmetric solution shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Border-collision-saddle-node-bif" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:limiting-behaviour" \end_inset \end_layout \end_inset \end_layout \end_inset Here, the interaction of the linear flow and the switched feedback cause the solution to intersect the points between which the fixed point switches. If \begin_inset Formula $\Omega$ \end_inset were to be decreased further, the solution would drop from \begin_inset Formula $\nu=5$ \end_inset to \begin_inset Formula $\nu=1$ \end_inset (and hence, the number of variables required to describe the state of the system drops from 7 to 3), but unlike the BCSN bifurcation above, this causes the feedback to change discontinuously, and so the solution ceases to exist. \end_layout \begin_layout Standard It is difficult to attached a label to this bifurcation. The described behaviour is, of course, completely impossible in a smooth system. However, it may be a generic bifurcation in systems which feature smooth flow and switching fixed points (caused by switched feedback, or possibly, switched forcing), and so we should consider extending the nomenclature to piecewise-smooth (PWS) systems. The observed behaviour is comparable to both a homoclinic bifurcation, where a limit cycle vanishes after colliding with a saddle, and a heteroclinic bifurcation, where a limit cycle vanishes after colliding with a pair of saddles at two or more points on its orbit \begin_inset CommandInset citation LatexCommand cite key "guckenheimer2013nonlinear,homburg2010homoclinic" literal "false" \end_inset . However, in this system we do not have saddle points; instead, we having a stable spiral which switches dynamically between two points located on the switching manifold itself. Also, while the solution \emph on intersects \emph default the fixed point, it would not be accurate to say that it \emph on collides \emph default with it in any classical sense. The solution reaches the point \begin_inset Formula $\left(0,1\right)$ \end_inset while spiralling into the point \begin_inset Formula $\left(0,-1\right)$ \end_inset , but the feedback switches precisely when it reaches \begin_inset Formula $\left(0,1\right)$ \end_inset , just in time for it to be the fixed point, then switches back to allow the solution to continue spiralling into \begin_inset Formula $\left(0,-1\right)$ \end_inset . This is a highly degenerate situation, but it is perhaps one in which a switching stable spiral might act like a pair of saddles. \end_layout \begin_layout Standard It might be reasonable to refer to the observed behaviour as a \emph on PWS heteroclinic bifurcation \emph default . While we compared the behaviour to both a homoclinic and a heteroclinic bifurcation, heteroclinic bifurcations occur generically in systems with symmetry. This system has symmetry, and it is reasonable to expect that breaking that symmetry would cause the limit cycle to intersect the stable spiral at only one point, which might be better described as a \emph on PWS homoclinic bifurcation \emph default . However, a significant amount of further work will be required to determine the validity of these deductions. \end_layout \begin_layout Standard \end_layout \begin_layout Section Conclusion \end_layout \begin_layout Standard In this chapter, we studied a piecewise-linear second-order delay-differential equation with delayed relay feedback. The system featured a complex bifurcation structure with extensive multistabili ty. We simulated the system analytically using an iterative map with variable time-step, and analysed the bifurcations of the system through Poincaré sections. We found that the stability of solutions are governed by smooth bifurcations, and the existence of solutions are governed by non-smooth bifurcations. \end_layout \begin_layout Standard There are two notably similar systems comparable to the one studied here. The first is an undamped harmonic oscillator with delayed relay feedback, derived from an optical model of pupil light reflex \begin_inset CommandInset citation LatexCommand cite key "longtin1990oscillatory,bayer1998oscillation,barton2006periodic" literal "false" \end_inset . In this system, solutions consist of sections of circles, instead of the sections of spirals we observe. A key difference in this model is that while the feedback is relies on \begin_inset Formula $x\left(t-\tau\right)$ \end_inset , it affects \begin_inset Formula $\dot{y}$ \end_inset , rather than \begin_inset Formula $\dot{x}$ \end_inset as in our model. This causes the effect of the change in feedback to act parallel to the switching manifold, rather than perpendicular as in our model. Bayer and an der Heiden \begin_inset CommandInset citation LatexCommand cite key "bayer1998oscillation" literal "false" \end_inset observed saddle-node and pitchfork bifurcations in this system, in addition to extensive multistability. Barton, Krauskopf and Wilson \begin_inset CommandInset citation LatexCommand cite key "barton2006periodic" literal "false" \end_inset studied a slightly smoothed approximation of this system, and observed Neimark-Sacker bifurcations in addition to those previously found. They also broke the symmetry of the system to unfold the bifurcations. \end_layout \begin_layout Standard Sieber \begin_inset CommandInset citation LatexCommand cite key "sieber2006dynamics" literal "false" \end_inset analysed an undamped oscillator with delayed relay feedback derived from an inverted pendulum. In this model, the feedback is comparable to our model, but with the addition of hysteresis. Hysteresis introduces a number of discontinuity-induced bifurcations not observed in our system, including corner-collision and grazing bifurcations. Sieber et al. \begin_inset CommandInset citation LatexCommand cite key "sieber2010dynamics" literal "false" \end_inset extended this analysis to a generic harmonic oscillator with an arbitrary switching manifold, rather than \begin_inset Formula $x=0$ \end_inset . The genericity of the system allowed it to yield a broader spectrum of smooth and non-smooth bifurcations. \end_layout \begin_layout Standard There is signficant further work required to fully understand both this system, and whether the observed phenomena are generic to a wider class of PWS systems. Furthermore, the scenario in which the PWS heteroclinic bifurcation is described is extremely unusual, and perhaps of lesser impact because the limit cycle is unstable. It would be desirable to determine a simple normal form in which a stable limit cycle undergoes a PWS heteroclinic bifurcation. It would also be worthwhile to break the symmetry of this system to see what happens to the PWS heteroclinic bifurcation. We expect that further work in this area would be of great interest. Based on the observed dynamics, this system is worthy of further study, and may grant insight into a wider class of PWS system. As such, we hope that this work will be of interest to a wide audience. \end_layout \begin_layout Chapter Dynamics of targeted ransomware negotiation \begin_inset CommandInset label LatexCommand label name "chap:Dynamics-of-targeted" \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash renewcommand{ \backslash leftmark}{Dynamics of targeted ransomware negotiation} \end_layout \end_inset \end_layout \begin_layout Standard The material in this chapter is largely reproduced from P. Ryan, J. Fokker, S. Healy and A. Amann, \emph on Dynamics of targeted ransomware negotiation \emph default , IEEE Access, 10:3283632844, 2022. \begin_inset CommandInset citation LatexCommand cite key "ryan2022dynamics" literal "false" \end_inset . The Analysis section has been extended with additional material considering the economic conditions under which targeted ransomware attacks are viable. My contribution to this work consisted of developing, defining and analysing the model, while my co-authors authors provided guidance on which key features of ransomware negotiation should be replicated, and which dynamical features should be retained or removed to produce a model which is relatively realistic and simple. \end_layout \begin_layout Section Abstract \end_layout \begin_layout Standard We consider how the development of targeted ransomware has affected the dynamics of ransomware negotiations to better understand how to respond to ransomware attacks. We construct a model of ransomware negotiations as an asymmetric non-cooperativ e two-player game. In particular, our model considers the investments that a cybercriminal must make in order to conduct a successful targeted ransomware attack. We demonstrate how imperfect information is a crucial feature for replicating observed real-world behaviour. Furthermore, we present optimal strategies for both the cybercriminal and the target, and demonstrate how imperfect information results in a non-trivial optimal strategy for the cybercriminal. \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard Computer security is a rapidly developing field, with new threats emerging and evolving constantly. As computer security providers develop their methods for detecting malware (malicious software), the cybercriminals behind the various strains of malware are forced to refine their techniques for avoiding detection, prompting further development from the computer security industry. As a result of the interaction between these competing agendas, problems in computer security can give rise to rich dynamical behaviour. By analysing these dynamics, we can provide insights that assist in understandi ng phenomena observed in computer security. We seek to provide insight on recent developments in ransomware by using game theory to explore the dynamics they have introduced. \end_layout \begin_layout Standard Ransomware is a type of malware designed to extort a ransom from the victim \begin_inset CommandInset citation LatexCommand citep key "kalaimannan2017influences,maigida2019systematic" literal "false" \end_inset , usually by denying the victim access to their computer or data until the ransom has been paid. In the past, ransomware relied on extorting a small amount of money from a large number of victims. The ransom itself would be fixed at a price low enough that nearly anyone could pay, but was typically non-negotiable, as negotiating a small ransom with many victims would not be worth the effort. In recent years, a new phenomenon known as \emph on targeted ransomware \emph default has emerged \begin_inset CommandInset citation LatexCommand citep key "beek2016targeted,bajpai2020dissecting" literal "false" \end_inset . Cybercriminals operating targeted ransomware specifically target large organisations, which can be extorted for significantly higher ransoms \begin_inset CommandInset citation LatexCommand citep key "coveware2020ransomware,zimba2019economic" literal "false" \end_inset . However, they are also likely to have a higher level of computer security, and so the cybercriminals are forced to make significant investment into breaching their security. Given the effort involved in breaching security, the cybercriminals will invest further in calculating the highest ransom they believe their target will pay. As a result of the larger sums of money at stake, cybercriminals are willing to negotiate the ransom demand to facilitate payment. We consider these negotiations to be a crucial feature in targeted ransomware, as their outcome has an enormous impact on both the cybercriminals and the targeted organisation. In this chapter, we develop a model of targeted ransomware negotiations based on game theory. Game theory is a branch of mathematics which studies strategic interactions between rational decision-makers \begin_inset CommandInset citation LatexCommand citep key "myerson2013game" literal "false" \end_inset . A game is a mathematical model with a clearly defined set of rules where two or more \emph on players \emph default make strategic decisions to influence the outcome of the game to their own personal benefit. By analysing the decisions available to players, optimal decision-making strategies can be determined which offer the best outcome for each player. Game theory can be applied to many real-life scenarios that involve competing interests by formulating a game as a mathematical abstraction of the given scenario. By analysing the decisions taken in the game logically, optimal strategies can be determined, granting insight into real-world behaviour. As such, game theory is highly applicable to fields such as ecology \begin_inset CommandInset citation LatexCommand citep key "brown1999ecology,mcgill2007evolutionary,smith1973logic" literal "false" \end_inset , economics \begin_inset CommandInset citation LatexCommand citep key "friedman1998economic,selten1990bounded,lukas2012earnouts,cerdeiro2017contagion" literal "false" \end_inset and politics \begin_inset CommandInset citation LatexCommand citep key "kydd1997game,ward1993game" literal "false" \end_inset . Very recently, ideas from game theory have been found to be useful in the study of ransomware. While all ransomware follows the same fundamental principles, there is sufficient variety in observed phenomena to merit a variety of models, such as defence and deterrence \begin_inset CommandInset citation LatexCommand citep key "caulfield2015optimizing,lindsay2015tipping,laszka2017economics,cartwright2019pay,hu2020optimal" literal "false" \end_inset , iterative negotiations \begin_inset CommandInset citation LatexCommand citep key "caporusso2018game" literal "false" \end_inset , price discrimination \begin_inset CommandInset citation LatexCommand citep key "hernandez2020economic" literal "false" \end_inset , incentive to return encrypted data \begin_inset CommandInset citation LatexCommand citep key "cartwright2019ransomware" literal "false" \end_inset , and sale of stolen data \begin_inset CommandInset citation LatexCommand citep key "li2021game" literal "false" \end_inset . Here, we examine how game theory can be applied to the growing threat of targeted ransomware. We construct a model of ransom negotiations as a game played between a cybercriminal and their target. The game focuses on the strategic behaviour and investments that the cybercrimi nal must commit to in order to implement a successful targeted ransomware attack. By analysing our model, we demonstrate how imperfect information is crucial for replicating observed real-world behaviour, and provide new insights into the real-world behaviour and strategy of cybercriminals operating strains of targeted ransomware. \end_layout \begin_layout Section Background \end_layout \begin_layout Standard In order to construct a model of targeted ransomware negotiations, we first consider the key features of targeted ransomware. On a technical level, one of the main differences between untargeted and targeted ransomware is how it spreads \begin_inset CommandInset citation LatexCommand citep key "panda2019targeted" literal "false" \end_inset . Untargeted ransomware is distributed indiscriminately, relying on victims with weak security to make a mistake that allows the ransomware to infect their computer. Such a strategy may be appealing to a cybercriminal, as it requires a low investment of effort to infect victims. However, this indiscriminate strategy is relatively easy to defend against. A potential victim can reduce their chances of being infected through practices such as maintaining good security, careful internet usage, and maintaining up-to-date data backups. Large organisations with valuable data are likely to have such practices implemented across a complex network of computers. In order to target such organisations, a cybercriminal must invest significant effort in circumventing their security and spreading their ransomware across the computer network. This typically involves multiple attack vectors, such as targeted phishing, remote desktop protocol (RDP), and searching for weak passwords \begin_inset CommandInset citation LatexCommand citep key "stahie2020ransomware" literal "false" \end_inset . The cybercriminal may spend days or weeks escalating their level of access to the target's network; in 2019, the mean dwell time (the duration a threat is present in a system before it is detected) of ransomware was 43 days \begin_inset CommandInset citation LatexCommand citep key "infocyte2019report" literal "false" \end_inset . Only when the cybercriminal has a level of access high enough to compromise the target's backups will they start encrypting files, ensuring that the only way to restore the data is by paying the ransom. While this process of circumventing security requires a large investment of time and effort, it enables the cybercriminal to extort organisations for ransoms far larger than previously possible \begin_inset CommandInset citation LatexCommand citep key "coalition2020cyber" literal "false" \end_inset . \end_layout \begin_layout Standard A crucial factor in any ransomware negotiation is the reliability of the cybercriminal. The victim's willingness to pay the ransom demand must be affected by the likelihood of getting their data back afterwards. There are two major reasons why a victim might get their data after paying. The first is that the cybercriminal chooses not to return the data, perhaps to avoid the cost incurred by doing so. This is not a sound business practice, as it results in a loss of perceived reliability that reduces a victim's willingness to pay, and hence, the cybercriminal's profits \begin_inset CommandInset citation LatexCommand citep key "cartwright2019ransomware" literal "false" \end_inset , which does not fit with the business-like stance that cybercriminals operating ransomware have adopted \begin_inset CommandInset citation LatexCommand citep key "asokan2020experts" literal "false" \end_inset . Therefore, we assume that our cybercriminal will always attempt to return the victim's data. But how does one unintentionally not return data? Modern ransomware operates by encrypting the victim's data using public-private key encryption, rendering the data unreadable until it has been decrypted \begin_inset CommandInset citation LatexCommand citep key "young1996cryptovirology" literal "false" \end_inset . In order to decrypt their data, the victim must pay the cybercriminal in exchange for the decryption key. As the encryption phase of the attack must proceed rapidly to avoid notice, flaws may arise during the process. If such flaws occur, then the decryption key will fail to decrypt the affected data. This results in some or all of the victim's files becoming permanently irretrievable, which is unlikely to be discovered until after the victim has paid the ransom. If a strain of ransomware is known to have a history of failure, the target's willingness to pay is reduced, and so the reliability of the cybercriminal's ransomware is an important factor in the negotiations. Ransomware strains can vary quite significantly in their reliability, depending on how heavily the cybercriminal invested in the development of the ransomware. They may even invest in providing \begin_inset Quotes eld \end_inset customer service \begin_inset Quotes erd \end_inset to their victims, walking them through the decryption process to further improve their image of reliability \begin_inset CommandInset citation LatexCommand citep key "ng2017malware" literal "false" \end_inset . \end_layout \begin_layout Standard The potential for negotiation adds a significant feature from game theory to the dynamics of targeted ransomware; information asymmetry. The two parties to the negotiation have different degrees of information about each other, information that is crucial to determining the outcome of the negotiation. The cybercriminal does not know exactly how much the target's data is worth to them. This is a significant factor, as the cybercriminal seeks to set the ransom as high as possible in order to justify the significant investment they make in their attack. Therefore, estimating the value of the target's files accurately is of great importance. To aid them in this, the cybercriminal can make use of both publicly available data, including shareholder's reports and valuations, and private data found on the target's computer network, such as up-to-date finances and business plans. However, uncovering and interpreting an organisation's private documents is not effortless, and so producing an accurate estimate for the value of the target's data requires further investment from the cybercriminal. While investing greater effort can lead to a more accurate estimate, it is highly unlikely that the cybercriminal will ever achieve perfect accuracy. Even so, we might expect the target to be at a distinct disadvantage in terms of information asymmetry. However, as a result of developments in the computer security industry in response to targeted ransomware, this is not entirely true. The potential for negotiation of large ransoms has led to the emergence of organisations that offer professional ransomware negotiation services \begin_inset CommandInset citation LatexCommand citep key "rundle2020ransomware" literal "false" \end_inset . The negotiators have experience in dealing with cybercriminals behind the various strains of ransomware, allowing them to negotiate effectively on the behalf of targeted organisations. The negotiators also offer specialised knowledge. This can include statistics such as the reliability of a given strain of malware, but it can also include less quantifiable information, such as how aggressively the cybercriminal behind a particular strain will negotiate. Operators of targeted ransomware are typically willing to reduce their demand in the interest of getting paid, but not all are equally receptive to negotiations. Some cybercriminals are likely to perceive a low counteroffer to their demand as an insult, and may react aggressively to punish the target. The severity of the reaction depends on how aggressive the cybercriminal is, which varies depending on individual and cultural factors. In extreme cases, they may even react by abandoning the negotiations, and their ransom, so that future targets will be less likely to negotiate. This aggressive behaviour is a double-edged sword; if the cybercriminal is not aggressive enough, then their targets will not pay them a large ransom. If they are too aggressive, their inclination to punish their targets for perceived insults will cost them ransoms. In order to be successful, the cybercriminal must balance their aggression with their investments in their strain of targeted ransomware. In the next section, we construct a model of ransomware negotiation that is based on these key features. \end_layout \begin_layout Section Modelling \end_layout \begin_layout Standard We propose to study the dynamics of targeted ransomware by modelling the negotiations as a two-player game. This approach was inspired by Selten's analysis of a two-player game modelling the interaction between a hostage taker and a hostage negotiator \begin_inset CommandInset citation LatexCommand citep key "selten1988simple" literal "true" \end_inset . Our two players are the attacker \begin_inset Formula $A$ \end_inset and the defender \begin_inset Formula $D$ \end_inset . Player \begin_inset Formula $A$ \end_inset is a cybercriminal (or group of cybercriminals) operating a strain of targeted ransomware. Player \begin_inset Formula $D$ \end_inset is an organisation targeted by player \begin_inset Formula $A$ \end_inset , assisted by a professional negotiator hired to negotiate the ransom. \end_layout \begin_layout Subsection Player \begin_inset Formula $A$ \end_inset 's investment \end_layout \begin_layout Standard As noted in the previous section, there are three areas in which player \begin_inset Formula $A$ \end_inset must invest in order to pull off a successful targeted attack: \end_layout \begin_layout Itemize The circumvention of player \begin_inset Formula $D$ \end_inset 's security; \end_layout \begin_layout Itemize the reliability of player \begin_inset Formula $A$ \end_inset 's ransomware; and \end_layout \begin_layout Itemize the estimation of the value of player \begin_inset Formula $D$ \end_inset 's data. \end_layout \begin_layout Standard We will not be considering player \begin_inset Formula $A$ \end_inset 's investment in circumventing player \begin_inset Formula $D$ \end_inset 's security here. While it is likely to be an important factor in the overall dynamics of the system, it has little bearing on the negotiations. Once player \begin_inset Formula $D$ \end_inset 's computer network has been infected, this investment only affects player \begin_inset Formula $A$ \end_inset 's net profit, and affects player \begin_inset Formula $D$ \end_inset not at all. However, the other two investments are very significant to the negotiations. \end_layout \begin_layout Standard By the time player \begin_inset Formula $A$ \end_inset launches their attack, they have already developed their strain of ransomware. In particular, they have invested in the reliability of their ransomware. Investing in reliability is important; if the decryption process is likely to fail, then player \begin_inset Formula $D$ \end_inset will not be willing to pay very much for the decryption key. This is a significant up-front development cost; it does not scale with the number of targets player \begin_inset Formula $A$ \end_inset attacks. For the sake of simplicity, we assume that player \begin_inset Formula $A$ \end_inset can amortize this investment over the targets that they will infect with their strain of ransomware. We refer to this investment as \begin_inset Formula $I_{\beta}$ \end_inset . As \begin_inset Formula $I_{\beta}$ \end_inset increases, so does the reliability of player \begin_inset Formula $A$ \end_inset 's ransomware. Let \begin_inset Formula $\beta$ \end_inset be the probability that the decryption key successfully decrypts data encrypted by the ransomware. We choose \begin_inset Formula $\beta$ \end_inset such that \begin_inset Formula \begin{equation} \beta=\frac{I_{\beta}}{I_{\beta}+I_{50}}\label{eq:beta} \end{equation} \end_inset where \begin_inset Formula $I_{50}$ \end_inset is the amount of investment required to achieve a reliability of \begin_inset Formula $50\%$ \end_inset , or \begin_inset Formula $\beta=0.5$ \end_inset . \begin_inset Formula $I_{50}$ \end_inset can be considered as an economic scaling factor, and determines the amount of development that player \begin_inset Formula $A$ \end_inset can purchase for a given investment. This choice of \begin_inset Formula $\beta$ \end_inset results in diminishing return for large \begin_inset Formula $I_{\beta}$ \end_inset , so that \begin_inset Formula $\beta\rightarrow1$ \end_inset as \begin_inset Formula $I_{\beta}\rightarrow\infty$ \end_inset , as shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:beta" \end_inset . Note that \begin_inset Formula $I_{50}$ \end_inset and \begin_inset Formula $I_{\beta}$ \end_inset are dimensionless, so that our model remains generally applicable. \begin_inset Formula $I_{50}=0.02$ \end_inset means that an investment of 2% of the value of the targeted files will yield a decryptor that is 50% reliable. While player \begin_inset Formula $D$ \end_inset doesn't know the value of \begin_inset Formula $I_{\beta}$ \end_inset , their negotiator can provide an estimate of \begin_inset Formula $\beta$ \end_inset from their experience of individual ransomware strains. \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../dynamics_of_targeted_ransomware_negotiation/fig_beta.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:beta" \end_inset \begin_inset Formula $\beta$ \end_inset , the probability of succesful decryption, depends on \begin_inset Formula $I_{\beta}$ \end_inset and \begin_inset Formula $I_{50}$ \end_inset . \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The final area in which player \begin_inset Formula $A$ \end_inset can invest is in their estimation of how much player \begin_inset Formula $D$ \end_inset 's data is worth. Without an accurate estimate, player \begin_inset Formula $A$ \end_inset is unlikely to choose an optimal ransom demand, and so investing in producing an accurate estimate is an important aspect of their strategy. We let \begin_inset Formula $x$ \end_inset be the value that player \begin_inset Formula $D$ \end_inset attaches to their encrypted data, and let \begin_inset Formula $\tilde{x}$ \end_inset be player \begin_inset Formula $A$ \end_inset 's estimate of \begin_inset Formula $x$ \end_inset . We refer to player \begin_inset Formula $A$ \end_inset 's investment in data value estimation as \begin_inset Formula $I_{\sigma}$ \end_inset . As \begin_inset Formula $I_{\sigma}$ \end_inset increases, so does the probability that \begin_inset Formula $\tilde{x}$ \end_inset will be close to \begin_inset Formula $x$ \end_inset . In this analysis, we choose to model \begin_inset Formula $\tilde{x}$ \end_inset as random variable following a \begin_inset Formula $\text{Lognormal\ensuremath{\left(\mu,\sigma^{2}\right)}}$ \end_inset distribution \begin_inset CommandInset citation LatexCommand citep key "crow1987lognormal" literal "false" \end_inset with probability density function \begin_inset Formula \begin{equation} f\left(\tilde{x},\mu,\sigma\right)=\frac{1}{\tilde{x}\sigma\sqrt{2\pi}}\exp\left(-\frac{\left(\ln\tilde{x}-\mu\right)^{2}}{2\sigma^{2}}\right)\label{eq:lognormal_pdf} \end{equation} \end_inset for \begin_inset Formula $\tilde{x}>0$ \end_inset and parameters \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \mu & =\ln x\nonumber \\ \sigma & =1-\frac{I_{\sigma}}{I_{50}+I_{\sigma}}\label{eq:mu_sigma} \end{align} \end_inset With the chosen parameters, \begin_inset Formula $\tilde{x}$ \end_inset has median \begin_inset Formula $x$ \end_inset . As \begin_inset Formula $I_{\sigma}\rightarrow\infty$ \end_inset , \begin_inset Formula $\sigma\rightarrow0$ \end_inset , the variance \begin_inset Formula $\left(e^{\sigma^{2}}-1\right)\left(e^{2\mu+\sigma^{2}}\right)\rightarrow0$ \end_inset and mean \begin_inset Formula $xe^{\frac{\sigma^{2}}{2}}\rightarrow x$ \end_inset . The Lognormal distribution has previously been used for modelling positive quantities that are determined by human behaviour \begin_inset CommandInset citation LatexCommand citep key "gualandi2019human" literal "false" \end_inset . As \begin_inset Formula $\tilde{x}$ \end_inset is player \begin_inset Formula $A$ \end_inset 's estimate of how much player \begin_inset Formula $D$ \end_inset values their data, we believe that this is an appropriate choice. Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:lognormal" plural "false" caps "false" noprefix "false" \end_inset shows the probability density function \begin_inset Formula $f\left(\tilde{x},\mu,\sigma\right)$ \end_inset for \begin_inset Formula $x=1$ \end_inset and varying levels of investment \begin_inset Formula $I_{\sigma}$ \end_inset . \begin_inset Formula $I_{\sigma}$ \end_inset is scaled similarly to \begin_inset Formula $I_{\beta}$ \end_inset . As \begin_inset Formula $I_{\sigma}$ \end_inset increases, the distribution narrows around \begin_inset Formula $x$ \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../dynamics_of_targeted_ransomware_negotiation/fig_lognormal.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:lognormal" \end_inset Probability density function \begin_inset Formula $f\left(\tilde{x},\mu,\sigma\right)$ \end_inset for \begin_inset Formula $x=1$ \end_inset , \begin_inset Formula $I_{50}=0.02$ \end_inset and varying levels of investment \begin_inset Formula $I_{\sigma}$ \end_inset . \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Negotiation \end_layout \begin_layout Standard Once player \begin_inset Formula $A$ \end_inset has infected player \begin_inset Formula $D$ \end_inset 's computer network with ransomware, the negotiation begins. Player \begin_inset Formula $A$ \end_inset issues a ransom demand \begin_inset Formula $R$ \end_inset , and player \begin_inset Formula $D$ \end_inset must decide how to respond. We now examine the negotiation process between player \begin_inset Formula $A$ \end_inset and player \begin_inset Formula $D$ \end_inset that occurs if player \begin_inset Formula $D$ \end_inset is willing to pay, but does not wish to pay the full demand. This is a time-sensitive issue, particularly for player \begin_inset Formula $D$ \end_inset . Player \begin_inset Formula $D$ \end_inset cannot conduct business while their data is encrypted, but they still incur costs, which can grow significant over a protracted negotiation. Player \begin_inset Formula $A$ \end_inset does not suffer such ongoing costs, but they have invested significant resources in the attack, and the longer the negotiation continues, the more time player \begin_inset Formula $D$ \end_inset has to consider their position. This makes a rapid negotiation process highly desirable, and so we make a simplification to the negotiation process and model it in the simplest way possible in the manner used by Selten \begin_inset CommandInset citation LatexCommand citep key "selten1988simple" literal "true" \end_inset . Player \begin_inset Formula $A$ \end_inset issues a ransom demand, player \begin_inset Formula $D$ \end_inset responds with a counteroffer \begin_inset Formula $C$ \end_inset , then player \begin_inset Formula $A$ \end_inset decides whether to accept \begin_inset Formula $C$ \end_inset and hand over the decryption key, or reject \begin_inset Formula $C$ \end_inset and abandon the negotiations. \end_layout \begin_layout Standard This is a substantially simplified description of the negotiation process and should not be taken literally. In reality, there may be a series of offers and counteroffers that take place over time. However, player \begin_inset Formula $D$ \end_inset has finite capital with which they can absorb the costs incurred by not being able to conduct business. A drawn out negotiation for a lower ransom may be more costly than a prompt negotiation for a higher ransom. \end_layout \begin_layout Standard Why would player \begin_inset Formula $A$ \end_inset ever reject \begin_inset Formula $C$ \end_inset ? In doing so, they lose both their potential earnings and waste any investment they've made in the attack, which is clearly an undesirable outcome. However, player \begin_inset Formula $A$ \end_inset must maintain their status as a threat; if they appear to be willing to accept low counteroffers, they will only receive low counteroffers. In order to maintain their credibility and their profits, player \begin_inset Formula $A$ \end_inset may punish player \begin_inset Formula $D$ \end_inset for making a low counteroffer. Therefore, we must expect that with a positive probability \begin_inset Formula $\alpha$ \end_inset , \begin_inset Formula $A$ \end_inset will perceive a counteroffer \begin_inset Formula $C0$ \end_inset is the aggression parameter of player \begin_inset Formula $A$ \end_inset , quantifying their tendency to perceive a low counteroffer as an affront and react aggressively. This aggressive reaction is different to that implemented by Selten \begin_inset CommandInset citation LatexCommand citep key "selten1988simple" literal "true" \end_inset , where \begin_inset Formula $\alpha=a\left(1-\frac{C}{R}\right)$ \end_inset and \begin_inset Formula $a\in\left[0,1\right]$ \end_inset so that \begin_inset Formula $\alpha\leq a\leq1$ \end_inset . The reason for this is that in Selten's game, the aggressive reaction of the hostage taker is to kill the hostage, while in our game, the aggressive reaction is merely to not decrypt data. It is reasonable to assume that a hostage taker, faced with having their ransom demand being disregarded, might still refrain from killing their hostage. However, a cybercriminal, divorced from the consequences of their actions, would have no reason not to react aggressively and abandon the negotiation. This choice of \begin_inset Formula $\alpha$ \end_inset allows for a wide range of behaviour from player \begin_inset Formula $A$ \end_inset , with very lenient negotiations for \begin_inset Formula $a<1$ \end_inset , scaling up to very aggressive negotiations as \begin_inset Formula $a$ \end_inset increases. Larger \begin_inset Formula $a$ \end_inset causes \begin_inset Formula $\alpha$ \end_inset to increase more rapidly as the difference between \begin_inset Formula $C$ \end_inset and \begin_inset Formula $R$ \end_inset increases, as shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:alpha" \end_inset . As with \begin_inset Formula $\beta$ \end_inset , player \begin_inset Formula $D$ \end_inset can estimate \begin_inset Formula $a$ \end_inset through the negotiator's experience of interacting with individual ransomware operators. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../dynamics_of_targeted_ransomware_negotiation/fig_alpha.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:alpha" \end_inset \begin_inset Formula $\alpha$ \end_inset , the probability of an aggressive reaction from player \begin_inset Formula $A$ \end_inset , depends on \begin_inset Formula $a$ \end_inset and the ratio of counteroffer to ransom demand \begin_inset Formula $\frac{C}{R}$ \end_inset . \end_layout \end_inset \end_layout \end_inset If player \begin_inset Formula $A$ \end_inset does not react aggressively to a counteroffer \begin_inset Formula $C \begin_inset Text \begin_layout Plain Layout Outcome \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Payoff \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Player \begin_inset Formula $A$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Player \begin_inset Formula $D$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Counteroffer rejected \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-I_{\beta}-I_{\sigma}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Counteroffer accepted, data decrypted successfully \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $C-I_{\beta}-I_{\sigma}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-C$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Counteroffer accepted, data not decrypted successfully \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $C-I_{\beta}-I_{\sigma}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-x-C$ \end_inset \end_layout \end_inset \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Summary of game rules \end_layout \begin_layout Standard The model variables are summarised in Table \begin_inset CommandInset ref LatexCommand ref reference "tab:variables" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset Float table wide true sideways true status open \begin_layout Plain Layout \noindent \align center \begin_inset Caption Standard \begin_layout Plain Layout Table of game variables with detail on the information asymmetry in the targeted ransomware negotiation game. \begin_inset CommandInset label LatexCommand label name "tab:variables" \end_inset \end_layout \end_inset \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Variable \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Description \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Known to player \begin_inset Formula $A$ \end_inset ? \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Known to player \begin_inset Formula $D$ \end_inset ? \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout True value of \begin_inset Formula $D$ \end_inset 's files \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout No \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tilde{x}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $A$ \end_inset 's estimate of \begin_inset Formula $x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout No \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $R$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $A$ \end_inset 's ransom demand \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $C$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $D$ \end_inset 's counteroffer \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $a$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $A$ \end_inset 's aggression \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $I_{\beta}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $A$ \end_inset 's investment in reliability \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout No \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $I_{\sigma}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $A$ \end_inset 's investment in estimating \begin_inset Formula $x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout No \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\alpha$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Probability that \begin_inset Formula $A$ \end_inset will react aggressively to \begin_inset Formula $C \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\beta$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Probability that the decryption key works \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\sigma$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Scale parameter of distribution of \begin_inset Formula $\tilde{x}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Yes \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout No \end_layout \end_inset \end_inset \end_layout \end_inset The rules of the game are summarised as follows: \end_layout \begin_layout Enumerate Player \begin_inset Formula $A$ \end_inset incurs cost \begin_inset Formula $I_{\beta}+I_{\sigma}$ \end_inset to infect player \begin_inset Formula $D$ \end_inset 's computer system and make a ransom demand \begin_inset Formula $R$ \end_inset . \end_layout \begin_layout Enumerate Player \begin_inset Formula $D$ \end_inset makes a counteroffer \begin_inset Formula $C$ \end_inset . \end_layout \begin_layout Enumerate Player \begin_inset Formula $A$ \end_inset rejects player \begin_inset Formula $D$ \end_inset 's counteroffer with probability \begin_inset Formula $\alpha$ \end_inset . \end_layout \begin_layout Enumerate If player \begin_inset Formula $A$ \end_inset does not reject the counteroffer, player \begin_inset Formula $A$ \end_inset receives the counteroffer \begin_inset Formula $C$ \end_inset and player \begin_inset Formula $D$ \end_inset receives the decryption key. \end_layout \begin_layout Enumerate Player \begin_inset Formula $D$ \end_inset 's data is successfully decrypted with probability \begin_inset Formula $\beta$ \end_inset . \end_layout \begin_layout Standard \begin_inset Separator plain \end_inset \end_layout \begin_layout Section Analysis \end_layout \begin_layout Subsection Optimal choice of \begin_inset Formula $C$ \end_inset \end_layout \begin_layout Standard In the subgame beginning with player \begin_inset Formula $D$ \end_inset 's choice of \begin_inset Formula $C$ \end_inset , player \begin_inset Formula $D$ \end_inset knows that making a counteroffer \begin_inset Formula $CR$ \end_inset , so they rationally chooses \begin_inset Formula $C$ \end_inset to maximize their expected utility \begin_inset Formula $U$ \end_inset : \begin_inset Formula \begin{equation} U=\begin{cases} -\left[R+\left(1-\beta\right)x\right] & C=R\\ -\left(1-\alpha\right)\left[C+\left(1-\beta\right)x\right]-\alpha x & CC_{max}$ \end_inset . By substituting from Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:alpha" \end_inset ), player \begin_inset Formula $D$ \end_inset 's expected utility for \begin_inset Formula $C\frac{a\beta x}{1+a}$ \end_inset ; for any \begin_inset Formula $R>\frac{a\beta x}{1+a}$ \end_inset , player \begin_inset Formula $D$ \end_inset 's optimal counteroffer is \begin_inset Formula $C=\frac{a\beta x}{1+a}$ \end_inset . \begin_inset Formula $\frac{\partial U}{\partial C}>0$ \end_inset for \begin_inset Formula $C<\frac{a\beta x}{1+a}$ \end_inset ; however, player \begin_inset Formula $D$ \end_inset will never make a counteroffer \begin_inset Formula $C>R$ \end_inset . If \begin_inset Formula $R<\frac{a\beta x}{1+a}$ \end_inset , player \begin_inset Formula $D$ \end_inset 's optimal counteroffer is \begin_inset Formula $C=R$ \end_inset . Thus, \begin_inset Formula $C_{max}=\frac{a\beta x}{1+a}$ \end_inset , yielding player \begin_inset Formula $D$ \end_inset 's optimal counteroffer \begin_inset Formula $\hat{C}$ \end_inset : \begin_inset Formula \begin{equation} \hat{C}=\begin{cases} R & R\leq\frac{a\beta x}{1+a}\\ \frac{a\beta x}{1+a} & R>\frac{a\beta x}{1+a} \end{cases}\label{eq:c_hat} \end{equation} \end_inset This scenario is demonstrated graphically in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:utility" \end_inset . Any choice of \begin_inset Formula $C_{max}\neq\frac{a\beta x}{1+a}$ \end_inset results in a drop in player \begin_inset Formula $D$ \end_inset 's expected utility. \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../dynamics_of_targeted_ransomware_negotiation/fig_utility.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:utility" \end_inset Player \begin_inset Formula $D$ \end_inset 's expected utility for varying \begin_inset Formula $R$ \end_inset and different values of \begin_inset Formula $C_{max}$ \end_inset where \begin_inset Formula $a=10$ \end_inset , \begin_inset Formula $I_{50}=0.02$ \end_inset and \begin_inset Formula $I_{\beta}=0.1$ \end_inset . \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Optimal choice of \begin_inset Formula $R$ \end_inset \end_layout \begin_layout Standard In the subgame beginning with player \begin_inset Formula $A$ \end_inset 's choice of \begin_inset Formula $R$ \end_inset , player \begin_inset Formula $A$ \end_inset knows that under rational decision-making, player \begin_inset Formula $D$ \end_inset will optimally make counteroffer \begin_inset Formula $\hat{C}$ \end_inset . Player \begin_inset Formula $A$ \end_inset rationally chooses \begin_inset Formula $R$ \end_inset to maximize their expected profit \begin_inset Formula $P$ \end_inset : \begin_inset Formula \begin{equation} P=\begin{cases} R-I_{\beta}-I_{\sigma} & R\leq C\\ \left(1-\alpha\right)C-I_{\beta}-I_{\sigma} & R>C \end{cases} \end{equation} \end_inset Substituting from Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:c_hat" \end_inset ) yields \begin_inset Formula \begin{equation} P=\begin{cases} R-I_{\beta}-I_{\sigma} & R\leq\frac{a\beta x}{1+a}\\ \left(\frac{\frac{a\beta x}{1+a}}{R}\right)^{a}\left(\frac{a\beta x}{1+a}\right)-I_{\beta}-I_{\sigma} & R>\frac{a\beta x}{1+a} \end{cases}\label{eq:p1} \end{equation} \end_inset By differentiating with respect to \begin_inset Formula $R$ \end_inset we find that \begin_inset Formula $\frac{\partial P}{\partial R}>0$ \end_inset for \begin_inset Formula $R<\frac{a\beta x}{1+a}$ \end_inset and \begin_inset Formula $\frac{\partial P}{\partial R}<0$ \end_inset for \begin_inset Formula $R>\frac{a\beta x}{1+a}$ \end_inset . Therefore, the optimal ransom demand \begin_inset Formula $\hat{R}=\frac{a\beta x}{1+a}$ \end_inset is the highest ransom that player \begin_inset Formula $D$ \end_inset is willing to pay. If player \begin_inset Formula $A$ \end_inset can reliably make demand \begin_inset Formula $\hat{R}$ \end_inset , player \begin_inset Formula $D$ \end_inset will always pay. Player \begin_inset Formula $A$ \end_inset will always make their maximum profit, and there is no risk of an aggressive reaction from player \begin_inset Formula $A$ \end_inset , trivialising the negotiation. Under such conditions, player \begin_inset Formula $A$ \end_inset 's profit is \begin_inset Formula $\frac{a\beta x}{1+a}-I_{\beta}-I_{\sigma}$ \end_inset . This would suggest that, in order to maximise their profit, player \begin_inset Formula $A$ \end_inset should be infinitely aggressive, rejecting any counteroffer even slightly lower than their demand (i.e. \begin_inset Formula $a\rightarrow\infty$ \end_inset ). \end_layout \begin_layout Standard Of course, this \begin_inset Quotes eld \end_inset ideal \begin_inset Quotes erd \end_inset scenario is unrealistic, as it ignores the often-significant effect of imperfect information \begin_inset CommandInset citation LatexCommand citep key "kreps1982reputation,barrachina2014entry,durkota2015approximate" literal "false" \end_inset . In reality, negotiations are not trivial affairs, and the risk of an aggressive reaction is always present. Player \begin_inset Formula $A$ \end_inset does not know \begin_inset Formula $x$ \end_inset , only \begin_inset Formula $\tilde{x}$ \end_inset , which, lacking any alternative, is what they use to calculate their ransom demand. Therefore, under optimal play while accounting for imperfect information, player \begin_inset Formula $A$ \end_inset 's ransom demand is \begin_inset Formula $R=\frac{a\beta\tilde{x}}{1+a}$ \end_inset . The potential for error in player \begin_inset Formula $A$ \end_inset 's estimate \begin_inset Formula $\tilde{x}$ \end_inset gives rise to the necessity of negotiations that may result in an aggressive reaction. We can illustrate this by substituting \begin_inset Formula $R=\frac{a\beta\tilde{x}}{1+a}$ \end_inset into Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:p1" plural "false" caps "false" noprefix "false" \end_inset ). Under optimal play, \begin_inset Formula \begin{equation} P=\begin{cases} \frac{a\beta\tilde{x}}{1+a}-I_{\beta}-I_{\sigma} & \frac{a\beta\tilde{x}}{1+a}\leq\frac{a\beta x}{1+a}\\ \left(\frac{\frac{a\beta x}{1+a}}{\frac{a\beta\tilde{x}}{1+a}}\right)^{a}\left(\frac{a\beta x}{1+a}\right)-I_{\beta}-I_{\sigma} & \frac{a\beta\tilde{x}}{1+a}>\frac{a\beta x}{1+a} \end{cases} \end{equation} \end_inset which factors to \begin_inset Formula \begin{equation} P=\frac{a\beta}{1+a}\left.\begin{cases} \tilde{x} & \tilde{x}\leq x\\ x\left(\frac{x}{\tilde{x}}\right)^{a} & \tilde{x}>x \end{cases}\right]-I_{\beta}-I_{\sigma}\label{eq:p2} \end{equation} \end_inset Thus, it is the potential for error in the estimate \begin_inset Formula $\tilde{x}$ \end_inset which prevents player \begin_inset Formula $A$ \end_inset from playing optimally. The effect of error in \begin_inset Formula $\tilde{x}$ \end_inset on player \begin_inset Formula $A$ \end_inset 's profit is shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Attacker's-net-profit" plural "false" caps "false" noprefix "false" \end_inset . Here, for investment levels are fixed at \begin_inset Formula $I_{\beta}=I_{\sigma}=0.1$ \end_inset , so that the maximum profit depends on \begin_inset Formula $a$ \end_inset . The effect of error differs depending on whether player \begin_inset Formula $A$ \end_inset underestimates or overestimates the value of the data. If they underestimate \begin_inset Formula $x$ \end_inset , then their profit decreases linearly with \begin_inset Formula $\tilde{x}$ \end_inset . If they overestimate \begin_inset Formula $x$ \end_inset , then the possibility of an aggressive reaction emerges, which increases with both \begin_inset Formula $a$ \end_inset , and the error in \begin_inset Formula $\tilde{x}$ \end_inset . High aggression might increase player \begin_inset Formula $A$ \end_inset 's capacity for demanding large ransoms, but at an increased risk of an aggressive reaction. \begin_inset Float figure placement h wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../dynamics_of_targeted_ransomware_negotiation/fig_attackers_profit_1.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Player \begin_inset Formula $A$ \end_inset 's expected profit as a function of \begin_inset Formula $\tilde{x}$ \end_inset for varying \begin_inset Formula $a$ \end_inset when \begin_inset Formula $x=1$ \end_inset , \begin_inset Formula $I_{50}=0.02$ \end_inset , and \begin_inset Formula $I_{\beta}=I_{\sigma}=0.1$ \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:Attacker's-net-profit" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Attacker's expected profit \end_layout \begin_layout Standard In order to further understand how error and aggression interact, we consider player \begin_inset Formula $A$ \end_inset 's expected net profit \begin_inset Formula $P$ \end_inset . First, let player \begin_inset Formula $A$ \end_inset 's gross profit be \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} GP\left(\tilde{x},x;a,\beta\right)=\frac{a\beta}{a+1}\begin{cases} \tilde{x} & \tilde{x}\leq x\\ x\left(\frac{x}{\tilde{x}}\right)^{a} & \tilde{x}>x \end{cases} \end{equation} \end_inset We let \begin_inset Formula $x$ \end_inset follow a distribution with probability density function \begin_inset Formula $g\left(x,M\right)$ \end_inset for \begin_inset Formula $x>0$ \end_inset and mean \begin_inset Formula $M$ \end_inset ; the shape of the distribution is unimportant in this case. With probability density functions \begin_inset Formula $g\left(x,M\right)$ \end_inset for \begin_inset Formula $x$ \end_inset and \begin_inset Formula $f\left(\tilde{x},\mu,\sigma\right)$ \end_inset for \begin_inset Formula $\tilde{x}$ \end_inset , Player \begin_inset Formula $A$ \end_inset 's net profit for a given combination of aggression and investments can be written as a convolution of \begin_inset Formula $x$ \end_inset and \begin_inset Formula $\tilde{x}$ \end_inset in double integral form. However, due to our choice of \begin_inset Formula $\alpha$ \end_inset and Lognormal \begin_inset Formula $\tilde{x}$ \end_inset , by making the substitution \begin_inset Formula $y=\frac{\tilde{x}}{x}$ \end_inset we can reduce the double integral to a single integral. \end_layout \begin_layout Standard \begin_inset Float table wide false sideways true status open \begin_layout Plain Layout \begin_inset Formula \begin{align} P\left(a,I_{\beta},I_{\sigma}\right) & =\int_{x=0}^{x=\infty}\int_{y=0}^{y=\infty}GP\left(xy,x;a,\beta\right)f\left(xy,\mu,\sigma^{2}\right)g\left(x,M\right)xdydx-I_{\beta}-I_{\sigma}\nonumber \\ & =\int_{x=0}^{x=\infty}\int_{y=0}^{y=\infty}\left(\frac{a\beta}{a+1}\right)\left.\begin{cases} xy & xy\leq x\\ x\left(\frac{1}{y}\right)^{a} & xy>x \end{cases}\right]\frac{1}{xy\sigma\sqrt{2\pi}}e^{-\frac{\left[\ln\left(xy\right)-\ln\left(x\right)\right]^{2}}{2\sigma^{2}}}g\left(x,M\right)xdydx-I_{\beta}-I_{\sigma}\nonumber \\ & =\left(\frac{a\beta}{a+1}\right)\int_{x=0}^{x=\infty}g\left(x,M\right)xdx\int_{y=0}^{y=\infty}\left.\begin{cases} y & y\leq1\\ y^{-a} & y>1 \end{cases}\right]\frac{1}{y\sigma\sqrt{2\pi}}e^{-\frac{\ln^{2}y}{2\sigma^{2}}}dy-I_{\beta}-I_{\sigma}\nonumber \\ & =\left(\frac{a\beta}{a+1}\right)M\left[\int_{y=0}^{y=1}y\frac{1}{y\sigma\sqrt{2\pi}}e^{-\frac{\ln^{2}y}{2\sigma^{2}}}dy+\int_{y=1}^{y=\infty}y^{-a}\frac{1}{y\sigma\sqrt{2\pi}}e^{-\frac{\ln^{2}y}{2\sigma^{2}}}dy\right]-I_{\beta}-I_{\sigma}\label{eq:profit_master} \end{align} \end_inset \end_layout \begin_layout Plain Layout \end_layout \end_inset \end_layout \begin_layout Standard By making this substitution, we can see that what appears to be a convolution of \begin_inset Formula $x$ \end_inset and \begin_inset Formula $\tilde{x}$ \end_inset depends merely on the ratio \begin_inset Formula $\frac{\tilde{x}}{x}$ \end_inset . Player \begin_inset Formula $A$ \end_inset 's strategy is consistent across all values of \begin_inset Formula $x$ \end_inset , so it is only the mean \begin_inset Formula $M$ \end_inset of the distribution of \begin_inset Formula $x$ \end_inset that remains in the final expression. In this form, we can clearly see where each element of player \begin_inset Formula $A$ \end_inset 's strategy \begin_inset Formula $\left(a,I_{\beta},I_{\sigma}\right)$ \end_inset comes into play. As aggression \begin_inset Formula $a$ \end_inset increases, the multiplicative term \begin_inset Formula $\frac{a}{a+1}$ \end_inset increases, but the second integral in the sum, where player \begin_inset Formula $A$ \end_inset has overestimated \begin_inset Formula $x$ \end_inset , converges to \begin_inset Formula $0$ \end_inset . Increasing \begin_inset Formula $I_{\beta}$ \end_inset increases costs, but leads to increased \begin_inset Formula $\beta$ \end_inset which may increase profit. Increasing \begin_inset Formula $I_{\sigma}$ \end_inset also increases costs, but narrows the distribution of \begin_inset Formula $\tilde{x}$ \end_inset around \begin_inset Formula $x$ \end_inset , allowing for greater aggression at decreased risk of aggressive reaction. Thus, through our choice of \begin_inset Formula $\alpha$ \end_inset and \begin_inset Formula $\tilde{x}$ \end_inset , we can more clearly demonstrate how player \begin_inset Formula $A$ \end_inset 's strategy depends on the interaction between the various elements of their strategy. The optimal counteroffer and expected net profit \begin_inset Formula $P$ \end_inset for parameters corresponding to sample strategies are shown in Table \begin_inset CommandInset ref LatexCommand ref reference "tab:strategies" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Caption Standard \begin_layout Plain Layout Model variables for optimal and various sub-optimal strategies. \begin_inset CommandInset label LatexCommand label name "tab:strategies" \end_inset \end_layout \end_inset \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Strategy Type \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(a,I_{\beta},I_{\sigma}\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $C$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $P$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Optimal \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(4.68,0.091,0.104\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.675$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.304$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Low Aggression \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(2.34,0.091,0.104\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.574$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.276$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout High Aggression \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(9.36,0.091,0.104\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.741$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.284$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Low Reliability \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(4.68,0.041,0.104\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.554$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.265$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout High Reliability \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(4.68,0.182,0.104\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.742$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.264$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Low Accuracy \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(4.68,0.091,0.052\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.675$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.283$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout High Accuracy \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(4.68,0.091,0.208\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.675$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.267$ \end_inset \end_layout \end_inset \end_inset \end_layout \end_inset Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:profit_master" plural "false" caps "false" noprefix "false" \end_inset shows \begin_inset Formula $P$ \end_inset for varying parameters \begin_inset Formula $a$ \end_inset , \begin_inset Formula $I_{\beta}$ \end_inset and \begin_inset Formula $I_{\sigma}$ \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../dynamics_of_targeted_ransomware_negotiation/fig_attackers_profit_numerical_integration_and_simulation.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Player \begin_inset Formula $A$ \end_inset 's profit for varying strategy parameters \begin_inset Formula $\left(a,I_{\beta},I_{\sigma}\right)$ \end_inset when \begin_inset Formula $I_{50}=0.02$ \end_inset . The plots on the left are the result of numerical integration, while the plots on the right are the result of averaging over \begin_inset Formula $n=10^{4}$ \end_inset simulations of the game with constant target data value \begin_inset Formula $x=1$ \end_inset . In each plot, the hidden parameter is set to its optimal value. The maximum mean profit achieved is marked by a black dot. The red curve marks where the mean profit is equal to 0. \begin_inset CommandInset label LatexCommand label name "fig:profit_master" \end_inset \end_layout \end_inset \end_layout \end_inset The left column shows results calculated via numerical integration of Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:profit_master" plural "false" caps "false" noprefix "false" \end_inset ) with \begin_inset Formula $M=1$ \end_inset , while the right column was calculated using an agent-based simulation, where results are averaged over \begin_inset Formula $n=10^{4}$ \end_inset simulations of the game with constant target data value \begin_inset Formula $x=1$ \end_inset . These figures demonstrate that player \begin_inset Formula $A$ \end_inset 's optimal strategy is \begin_inset Formula $\left(a,I_{\beta},I_{\sigma}\right)=\left(4.68,0.091,0.104\right)$ \end_inset , rather than the naive maximal aggression \begin_inset Formula $a\rightarrow\infty$ \end_inset noted previously; in order to realise their potential profits, the attacker must be willing to negotiate. \end_layout \begin_layout Standard We also observe that the results derived from agent-based simulation are quite noisy, despite being averaged over \begin_inset Formula $n=10^{4}$ \end_inset simulations. While increasing \begin_inset Formula $n$ \end_inset would reduce noise, this would come at the cost of increasing computation time. In Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:profit_err_time" plural "false" caps "false" noprefix "false" \end_inset , we consider how agent-based simulation compares to numerical integration as we vary \begin_inset Formula $n$ \end_inset with parameters \begin_inset Formula $\left(a,I_{\beta},I_{\sigma}\right)=\left(4.68,0.091,0.104\right)$ \end_inset . To avoid confusion, let \begin_inset Formula $P_{\infty}$ \end_inset be \begin_inset Formula $P$ \end_inset calculated from numerical integration using the Python function \begin_inset Formula $\texttt{scipy.integrate.quad}$ \end_inset \begin_inset CommandInset citation LatexCommand citep key "piessens2012quadpack,jones2001scipy" literal "false" \end_inset , and let \begin_inset Formula $P_{n}$ \end_inset be \begin_inset Formula $P$ \end_inset calculated from averaging over \begin_inset Formula $n$ \end_inset simulations. The estimated absolute error of \begin_inset Formula $P_{\infty}$ \end_inset is of order \begin_inset Formula $O\left(10^{-11}\right)$ \end_inset . In Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:profit_err_time" plural "false" caps "false" noprefix "false" \end_inset (a) and Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:profit_err_time" plural "false" caps "false" noprefix "false" \end_inset (b), we observe that the distribution of \begin_inset Formula $P_{n}$ \end_inset is well approximated by a Normal distribution, as predicted by the Central Limit Theorem (CLT) \begin_inset CommandInset citation LatexCommand citep key "casella1990statistical" literal "false" \end_inset , and that the standard deviation decreases (and the distribution narrows) as \begin_inset Formula $n$ \end_inset increases. In Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:profit_err_time" plural "false" caps "false" noprefix "false" \end_inset (c), we plot the standard deviation of \begin_inset Formula $P_{n}$ \end_inset for varying \begin_inset Formula $n$ \end_inset on a log-log scale, where the relationship between the two is linear with a slope of \begin_inset Formula $-\frac{1}{2}$ \end_inset , so that the standard deviation is proportional to \begin_inset Formula $n^{-\frac{1}{2}}$ \end_inset , which is again consistent with the CLT. In theory, we can simply increase \begin_inset Formula $n$ \end_inset until the standard deviation of \begin_inset Formula $P_{n}$ \end_inset is of lower order as the maximum error of \begin_inset Formula $P_{\infty}$ \end_inset . However in practice, reducing the standard deviation of \begin_inset Formula $P_{n}$ \end_inset to order \begin_inset Formula $O\left(10^{-3}\right)$ \end_inset takes more computation time than numerically integrating \begin_inset Formula $P_{\infty}$ \end_inset to within an absolute error order \begin_inset Formula $O\left(10^{-11}\right)$ \end_inset , as shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:profit_err_time" plural "false" caps "false" noprefix "false" \end_inset (d). Therefore, precisely estimating the attacker's expected net profit using numerical integration is far more efficient than using agent-based simulation, which justifies our efforts to reduce the problem to numerical integration. \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../dynamics_of_targeted_ransomware_negotiation/fig_simulation_vs_integration_error_time.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Comparison of error and computation time of the attacker's average net profit \begin_inset Formula $P$ \end_inset between numerical integration ( \begin_inset Formula $P_{\infty}$ \end_inset ) and agent-based simulation ( \begin_inset Formula $P_{n}$ \end_inset ) as the number of simulations \begin_inset Formula $n$ \end_inset is varied. \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none In (a) and (b), a Normal probability density function (red) is fitted to the distribution of \begin_inset Formula $P_{n}$ \end_inset (blue) for two values of \begin_inset Formula $n$ \end_inset . (c) shows the standard deviation of \begin_inset Formula $P$ \end_inset for varying \begin_inset Formula $n$ \end_inset . (d) shows the computation time of \begin_inset Formula $P_{n}$ \end_inset for varying \begin_inset Formula $n$ \end_inset (blue) and \begin_inset Formula $P_{\infty}$ \end_inset (dotted red). \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit \begin_inset CommandInset label LatexCommand label name "fig:profit_err_time" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Section Viability of targeted ransomware under varying economic conditions \end_layout \begin_layout Standard Thus far, we have neglected \begin_inset Formula $I_{50}$ \end_inset , the parameter that reflects economic conditions for the attacker, setting the scale for the amount of investment needed by the attacker to achieve a certain level of precision in decryptor reliability and target data value estimation. Now that we have considered how the attacker's profit varies with their chosen strategy, we can consider how their profit under optimal strategy varies with \begin_inset Formula $I_{50}$ \end_inset . Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:I50_optimal_strategy" plural "false" caps "false" noprefix "false" \end_inset shows the attacker's optimal strategy and expected profit under that strategy for for varying \begin_inset Formula $I_{50}$ \end_inset and fixed \begin_inset Formula $x=1$ \end_inset , so that as \begin_inset Formula $I_{50}$ \end_inset increases, the cost of precision and estimation increases relative to the value of the target's data. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../dynamics_of_targeted_ransomware_negotiation/fig_I50_dependence_optimal_strategy.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Attacker's optimal strategy and expected profit under that strategy for varying \begin_inset Formula $I_{50}$ \end_inset and fixed \begin_inset Formula $x=1$ \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:I50_optimal_strategy" \end_inset \end_layout \end_inset \end_layout \end_inset As \begin_inset Formula $I_{50}$ \end_inset increases, the attacker's expected profit decreases towards \begin_inset Formula $0$ \end_inset , as expected. The optimal strategy on the other hand, undergoes an interesting transition. Firstly, for small \begin_inset Formula $I_{50}\lesssim0.029$ \end_inset , \begin_inset Formula $I_{\sigma}>I_{\beta}$ \end_inset , indicating that when costs are extremely low relative to the value of the files, the attacker should focus on value estimation over reliability. \begin_inset Formula $I_{50}=0.02$ \end_inset , the value of \begin_inset Formula $I_{50}$ \end_inset used throughout our analysis, lies in this domain. For \begin_inset Formula $I_{50}\gtrsim0.029$ \end_inset , \begin_inset Formula $I_{\sigma}0$ \end_inset ; as \begin_inset Formula $\lambda_{U}$ \end_inset increases, so does the expected number of times the item is used in time \begin_inset Formula $T$ \end_inset . \end_layout \begin_layout Enumerate When the endpoint uses the item, it sends a query to the server, unless it has already queried the item in the preceding time interval of length \begin_inset Formula $\tau\geq0$ \end_inset , in which case the information is already cached on the endpoint and there is no need to query the server. Let \begin_inset Formula $Q=\left\{ q_{i}\right\} _{i=1}^{N_{Q}}$ \end_inset be the set of times at which the endpoint queries the server; then \begin_inset Formula $Q\subseteq U$ \end_inset and \begin_inset Formula $N_{Q}\leq N_{U}$ \end_inset . \end_layout \begin_layout Enumerate The server observes \series bold \begin_inset Formula $Q$ \end_inset \series default , but it applies \emph on simple random sampling \emph default (defined shortly) to avoid storing excessive amounts of data. Each query is recorded with probability \begin_inset Formula $\pi\in\left[0,1\right]$ \end_inset . Let \begin_inset Formula $R=\left\{ r_{i}\right\} _{i=1}^{N_{R}}$ \end_inset be the set of times at which a query is recorded; then \begin_inset Formula $R\subseteq Q$ \end_inset and \begin_inset Formula $N_{R}\leq N_{Q}$ \end_inset . \end_layout \begin_layout Standard These sets and set cardinalities evolve stochastically with time \begin_inset Formula $t$ \end_inset . Formally, \begin_inset Formula $U\left(t\right)$ \end_inset , \begin_inset Formula $Q\left(t\right)$ \end_inset and \begin_inset Formula $R\left(t\right)$ \end_inset are random processes, and their final states \begin_inset Formula $U=U\left(T\right)$ \end_inset , \begin_inset Formula $Q=Q\left(T\right)$ \end_inset and \begin_inset Formula $R=R\left(T\right)$ \end_inset are \emph on random variables \emph default ; the same treatment applies to \begin_inset Formula $N_{U}\left(t\right)$ \end_inset , \begin_inset Formula $N_{Q}\left(t\right)$ \end_inset and \begin_inset Formula $N_{R}\left(t\right)$ \end_inset . \end_layout \begin_layout Paragraph Simple random sampling \end_layout \begin_layout Standard The type of sampling employed here is referred to as simple random sampling \begin_inset CommandInset citation LatexCommand cite key "casella1990statistical" literal "false" \end_inset with probability \begin_inset Formula $\pi$ \end_inset . A simple explanation is that each query in \begin_inset Formula $Q$ \end_inset is individually added to \begin_inset Formula $R$ \end_inset with probability \begin_inset Formula $\pi$ \end_inset . More formally, for \begin_inset Formula $r\in R$ \end_inset and \begin_inset Formula $q\in Q$ \end_inset , \end_layout \begin_layout Enumerate The probability that a specific query \begin_inset Formula $q_{i}$ \end_inset is recorded is \begin_inset Formula \begin{equation} P\left[q_{i}\in R\right]=\pi \end{equation} \end_inset \end_layout \begin_layout Enumerate Each query \begin_inset Formula $q_{i}$ \end_inset can only be recorded once; \begin_inset Formula \begin{equation} P\left[r_{j}=q_{i}|r_{k}=q_{i}\right]=0 \end{equation} \end_inset if \begin_inset Formula $j\neq k$ \end_inset . \end_layout \begin_layout Enumerate The probability that a query \begin_inset Formula $q_{i}$ \end_inset is in \begin_inset Formula $R$ \end_inset is independent of whether any other query \begin_inset Formula $q_{l}$ \end_inset is in \begin_inset Formula $R$ \end_inset ; \begin_inset Formula \begin{equation} P\left[r_{j}=q_{i}|r_{k}=q_{l}\right]=\frac{1}{N_{Q}-1} \end{equation} \end_inset if \begin_inset Formula $j\neq k$ \end_inset and \begin_inset Formula $i\neq l$ \end_inset . If query \begin_inset Formula $q_{l}$ \end_inset is already known to be in record \begin_inset Formula $r_{k}$ \end_inset , then record \begin_inset Formula $r_{j}$ \end_inset can be any of the other \begin_inset Formula $N_{Q}-1$ \end_inset queries independently. \end_layout \begin_layout Standard The model has three parameters; data item usage rate \begin_inset Formula $\lambda_{U}$ \end_inset , caching time \begin_inset Formula $\tau$ \end_inset and sampling rate \begin_inset Formula $\pi$ \end_inset . From the perspective of the server, the system produces observable data \begin_inset Formula $Q$ \end_inset . The server chooses \begin_inset Formula $\pi$ \end_inset to reduce the cost of storing query data, and chooses \begin_inset Formula $\tau$ \end_inset as part of its strategy to reduce query volume through caching. To estimate how effective this strategy is, the server would like to estimate how many times an item is used while it is cached. To enable this, we would like to determine a method for estimating \begin_inset Formula $\lambda_{U}$ \end_inset by defining an estimator \begin_inset Formula $\hat{\lambda}_{U}$ \end_inset as a function of \begin_inset Formula $Q$ \end_inset , \begin_inset Formula $\tau$ \end_inset and \begin_inset Formula $\pi$ \end_inset \begin_inset CommandInset citation LatexCommand cite key "casella1990statistical,hogg1995introduction" literal "false" \end_inset . \end_layout \begin_layout Subsection Poisson process of item usage \end_layout \begin_layout Standard We assume that \begin_inset Formula $U\left(t\right)$ \end_inset is generated according to a Poisson process \begin_inset CommandInset citation LatexCommand cite key "gardiner2009stochastic,bickel2001mathematical,casella1990statistical" literal "false" \end_inset with constant rate \begin_inset Formula $\lambda_{U}>0$ \end_inset , such that \begin_inset Formula $N_{U}\left(t\right)$ \end_inset is a \emph on counting process; \emph default a non-decreasing random walk on the non-negative integers \begin_inset Formula $\left\{ 0\right\} \cup\mathbb{Z}^{+}$ \end_inset . Let \begin_inset Formula $P_{k}\left(t\right)=P\left[N_{U}\left(t\right)=k\right]$ \end_inset , then the random walk has transition probabilities \begin_inset Formula \begin{align} \frac{d}{dt}P_{0}\left(t\right) & =-\lambda_{U}P_{0}\left(t\right)\\ \frac{d}{dt}P_{k+1}\left(t\right) & =\lambda_{U}P_{k}\left(t\right)-\lambda_{U}P_{k+1}\left(t\right)\,\,k>0\\ P_{k}\left(0\right) & =\begin{cases} 1 & k=0\\ 0 & k>0 \end{cases} \end{align} \end_inset The Poisson process is one of the simplest stochastic processes for modelling a counting process, and has been widely used across diverse disciplines including astronomy \begin_inset CommandInset citation LatexCommand cite key "babu1996spatial,stoica2014spatial,scargle2003ch,alfaro2017search,tempel2016bisous,kovcivsvcak2023modeling" literal "false" \end_inset , biology \begin_inset CommandInset citation LatexCommand cite key "gosztolai2020cellular,codling2008random,othmer1988models" literal "false" \end_inset and communication networks \begin_inset CommandInset citation LatexCommand cite key "haenggi2009stochastic,heath2013modeling,hmamouche2021new" literal "false" \end_inset . If \begin_inset Formula $U\left(t\right)$ \end_inset evolves according to Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:poisson_process0" plural "false" caps "false" noprefix "false" \end_inset ), then \begin_inset Formula $N_{U}\left(T\right)$ \end_inset follows a Poisson distribution \begin_inset CommandInset citation LatexCommand cite key "gardiner2009stochastic" literal "false" \end_inset with rate \begin_inset Formula $\lambda_{U}T$ \end_inset \begin_inset Formula \begin{equation} N_{U}\left(T\right)\sim\text{Poi}\left(\lambda_{U}T\right) \end{equation} \end_inset such that for constant \begin_inset Formula $t=T$ \end_inset , the \emph on probability mass function \emph default is \begin_inset Formula \begin{equation} P\left[N_{U}\left(T\right)=k\right]=\frac{\left(\lambda_{U}T\right)^{k}e^{-\lambda_{U}T}}{k!} \end{equation} \end_inset (Derived in Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Derivations-of-Poisson" plural "false" caps "false" noprefix "false" \end_inset ). The expected value of \begin_inset Formula $N_{U}\left(T\right)$ \end_inset is \begin_inset Formula \begin{equation} \left\langle N_{U}\left(T\right)\right\rangle =\lambda_{U}T\label{eq:poisson_expected_value} \end{equation} \end_inset (Derived in Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Exp_value_poisson" plural "false" caps "false" noprefix "false" \end_inset ) which is the rate per unit time multiplied by elapsed time. \end_layout \begin_layout Standard An important property of the Poisson process which will used throughout this chapter is the distribution of the \emph on inter-event \emph default \emph on times \emph default . If \begin_inset Formula $u_{i}\in U\left(t\right)$ \end_inset are item usage times such that the \begin_inset Formula $u_{i}$ \end_inset are ordered by the index \begin_inset Formula $i$ \end_inset ; that is, \begin_inset Formula \begin{equation} i\tau \end{cases} \end{equation} \end_inset The expected value of \begin_inset Formula $\overline{q}_{i}$ \end_inset is \begin_inset Formula \begin{equation} \left\langle \overline{q}_{i}\right\rangle =\frac{1}{\lambda_{U}}+\tau \end{equation} \end_inset (see Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Exp_shifted_exponential" plural "false" caps "false" noprefix "false" \end_inset ) and hence, the expected number of queries is the total time divided by the expected inter-query time \begin_inset Formula \begin{equation} \left\langle N_{Q}\left(T\right)\right\rangle =\frac{T}{\left\langle \overline{q}_{i}\right\rangle }=\frac{T}{\frac{1}{\lambda_{U}}+\tau}\label{eq:expected_number_queries} \end{equation} \end_inset Finally, the expected number of records is given by \begin_inset Formula \begin{align} \left\langle N_{R}\left(T\right)\right\rangle & =\pi\left\langle N_{Q}\left(T\right)\right\rangle \label{eq:expected_number_record} \end{align} \end_inset We define the rate of accumulation of records as \begin_inset Formula \begin{equation} \lambda_{R}=\lim_{T\rightarrow\infty}\frac{\left\langle N_{R}\left(T\right)\right\rangle }{T} \end{equation} \end_inset which is estimated by \begin_inset Formula \begin{equation} \hat{\lambda}_{R}=\frac{\left\langle N_{R}\left(T\right)\right\rangle }{T}\label{eq:record_rate_estimator} \end{equation} \end_inset Combining Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:expected_number_queries" plural "false" caps "false" noprefix "false" \end_inset ), Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:expected_number_record" plural "false" caps "false" noprefix "false" \end_inset ) and Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:record_rate_estimator" plural "false" caps "false" noprefix "false" \end_inset ) we derive an estimator \begin_inset Formula $\hat{\lambda}_{U}$ \end_inset \begin_inset Formula \begin{align} \hat{\lambda}_{R} & =\frac{\left\langle N_{R}\left(T\right)\right\rangle }{T}=\frac{\pi\left\langle N_{Q}\left(T\right)\right\rangle }{T}=\frac{\pi}{\frac{1}{\lambda_{U}}+\tau} \end{align} \end_inset Rearranging, we derive an estimator \begin_inset Formula $\hat{\lambda}_{U}$ \end_inset \begin_inset Formula \begin{equation} \hat{\lambda}_{U}=\frac{1}{\frac{\pi}{\hat{\lambda}_{R}}-\tau}=\frac{1}{\frac{\pi T}{N_{R}\left(T\right)}-\tau}\label{eq:lambda_hat} \end{equation} \end_inset In Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:observed_lambda_vs_tau" plural "false" caps "false" noprefix "false" \end_inset , we simulate the system for \begin_inset Formula $T=1000$ \end_inset , and plot \begin_inset Formula $\hat{\lambda}_{R}$ \end_inset for varying parameters \begin_inset Formula $\lambda_{U}$ \end_inset , \begin_inset Formula $\tau$ \end_inset and \begin_inset Formula $\pi$ \end_inset , showing that our analytic result matches our simulation. \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/fig_observed_lambda_vs_tau.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout We show experimentally that the observed query rate is \begin_inset Formula $\lambda_{R}=\frac{\lambda_{U}\pi}{1+\tau\lambda_{U}}$ \end_inset . Here \begin_inset Formula $\lambda_{U}=10$ \end_inset and \begin_inset Formula $T=1000$ \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:observed_lambda_vs_tau" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Estimation of query volume reduction due to caching \end_layout \begin_layout Standard We now apply our estimate for \begin_inset Formula $\hat{\lambda}$ \end_inset derived above to estimate the reduction in query volume due to caching for a single endpoint and data item, and hence across the entire data network. Let \begin_inset Formula $E=\left\{ e_{i}\right\} _{i=1}^{N_{E}}$ \end_inset be the set of endpoints, and let \begin_inset Formula $D=\left\{ d_{j}\right\} _{j=1}^{N_{D}}$ \end_inset be the set of data items. There are \begin_inset Formula $N_{R}^{ij}\left(T\right)$ \end_inset records of endpoint \begin_inset Formula $e_{i}$ \end_inset querying data item \begin_inset Formula $d_{j}$ \end_inset after time \begin_inset Formula $T$ \end_inset . Hence, the item usage rate is estimated as \begin_inset Formula \begin{equation} \hat{\lambda}_{ij}=\frac{1}{\frac{\pi T}{N_{R}^{ij}\left(T\right)}-\tau} \end{equation} \end_inset The expected number of times that \begin_inset Formula $e_{i}$ \end_inset queries \begin_inset Formula $d_{j}$ \end_inset is \begin_inset Formula \begin{align} \left\langle N_{Q}^{ij}\left(T\right)\right\rangle & =\frac{1}{\pi}N_{R}^{ij}\left(T\right) \end{align} \end_inset Therefore, the expected aggregate time during which \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset Formula $d_{j}$ \end_inset was cached on \begin_inset Formula $e_{i}$ \end_inset i \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit s \begin_inset Formula $\frac{\tau}{\pi}N_{R}^{ij}\left(T\right)$ \end_inset , during which it was still being used with estimated rate \begin_inset Formula $\hat{\lambda}_{ij}$ \end_inset . Using the additive property of Poisson random variables \begin_inset CommandInset citation LatexCommand cite key "bickel2001mathematical" literal "false" \end_inset (see Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Sum-of-Poisson" plural "false" caps "false" noprefix "false" \end_inset ) the number of times the item was used while cached is \begin_inset Formula \begin{equation} \sum_{k=1}^{\frac{1}{\pi}N_{R}^{ij}\left(T\right)}\text{Poi}\left(\hat{\lambda}_{ij}\tau\right)=\text{Poi}\left(\hat{\lambda}_{ij}\frac{\tau}{\pi}N_{R}^{ij}\left(T\right)\right) \end{equation} \end_inset so that the expected number of queries prevented over time interval \begin_inset Formula $T$ \end_inset is \begin_inset Formula \begin{equation} P_{ij}\left(T\right)=\hat{\lambda}_{ij}\frac{\tau}{\pi}N_{R}^{ij}\left(T\right) \end{equation} \end_inset By summing over all endpoints and items, we estimate the expected total number of queries prevented by caching as \begin_inset Formula \begin{equation} P\left(T\right)=\frac{\tau}{\pi}\sum_{i,j}\hat{\lambda}_{ij}N_{R}^{ij}\left(T\right) \end{equation} \end_inset This estimate provides a useful metric for how effective a caching system is at preventing queries. However, it does not take into account whether there is a cost associated in reducing the number of queries through caching. In section \begin_inset CommandInset ref LatexCommand ref reference "sec:Optimal-TTL-for" plural "false" caps "false" noprefix "false" \end_inset , we consider a common scenario where reducing queries has its own cost. \end_layout \begin_layout Section Optimal TTL for caching volatile data \begin_inset CommandInset label LatexCommand label name "sec:Optimal-TTL-for" \end_inset \end_layout \begin_layout Standard In section \begin_inset CommandInset ref LatexCommand ref reference "sec:Poisson-process-model" plural "false" caps "false" noprefix "false" \end_inset we consider some of the complexities involved in estimating the rate of data item usage on an endpoint. Now we consider how such an estimate may be put to use in estimating the optimal TTL. \end_layout \begin_layout Standard As before, consider a single data item. The item is used on the endpoint according to a Poisson process with rate \begin_inset Formula $\lambda_{U}$ \end_inset ; if the item is used while it is not cached, the endpoint queries the server for the data item and then caches the item on the endpoint for time \begin_inset Formula $\tau$ \end_inset . However, this data item is also \emph on volatile \emph default ; it changes on the server according to some random process (presumed to be a Poisson process) with rate \begin_inset Formula $\mu$ \end_inset . If the item changes on the server while cached on an endpoint, it does not change on the endpoint and so becomes \emph on outdated \emph default . An endpoint using outdated data is a highly undesirable drop in service quality, and so there is a cost associated with an outdated item being used which is greater than the cost of the server responding to a query for a data item every time it is needed. Hence, the cost of a data item being used varies dynamically over time, as visualised in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:dynamic_usage_cost" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/fig_dynamic_model_diagram.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Sketch of dynamically varying data usage cost. When a data item is queried (blue circle), the usage cost for later queries drops for an interval \begin_inset Formula $\tau$ \end_inset while the item is cached, but if the data changes (red triangle) in that interval, the usage cost is greatly increased until the item leaves the cache. \begin_inset CommandInset label LatexCommand label name "fig:dynamic_usage_cost" \end_inset \end_layout \end_inset \end_layout \end_inset When a data item is queried (blue circle), the usage cost for later queries drops for an interval \begin_inset Formula $\tau$ \end_inset while the item is cached, but if the data changes (red triangle) in that interval, the usage cost is greatly increased until the item leaves the cache. Hence, the usage cost is piecewise-constant and varies dynamically based on the history of the item in the last \begin_inset Formula $\tau$ \end_inset time. \end_layout \begin_layout Standard As the cost of using outdated data items is relatively high, we will focus on scenarios where it is also relatively uncommon. Hence, we assume that \begin_inset Formula $\mu\tau\ll\lambda_{U}\tau$ \end_inset ; so that the expected number of times that a cached data item becomes outdated is much less than the expected number of times that a cached data item is used. Unlike estimating \begin_inset Formula $\lambda_{U}$ \end_inset , it is relatively straightforward for the server to estimate \begin_inset Formula $\mu$ \end_inset , as data item usage takes place at the endpoint, while data item changes takes place on the server. If a data item changes according to a Poisson process, then the server need only keep track of the number of times that the item has changed, and the interval of time the item has existed on the server, to estimate rate at which changes occur. \end_layout \begin_layout Standard For given \begin_inset Formula $\mu$ \end_inset and \begin_inset Formula $\lambda_{U}$ \end_inset , there is an optimal choice of \begin_inset Formula $\tau$ \end_inset which minimises the combined cost of queries and outdated data usage. In this section, we consider a model of item usage and item volatility which enables us to calculate the optimal choice of \begin_inset Formula $\tau$ \end_inset when the rate of data item change \begin_inset Formula $\mu$ \end_inset is small. \end_layout \begin_layout Subsection Rate of outdated data usage \end_layout \begin_layout Standard We assume that data items are used according to a Poisson process with rate \begin_inset Formula $\lambda_{U}$ \end_inset as in the previous section, and we assume that data items change according to a Poisson process with constant rate \begin_inset Formula $\mu$ \end_inset . Furthermore, we assume that the probability that a data item changes, and then changes back to its previous state, with a time interval of length \begin_inset Formula $\tau$ \end_inset is negligible. This is the case when \begin_inset Formula $\mu$ \end_inset is small, or potentially when the state space of the data item is large. We assume that \begin_inset Formula $\mu$ \end_inset is small; if \begin_inset Formula $\mu$ \end_inset is large, and using outdated data items is undesirable, then caching would not be an appropriate data management method. If a data item changes while cached, we say that the cached data item is now outdated. \end_layout \begin_layout Standard Consider a data item which is cached on the endpoint at time \begin_inset Formula $t_{0}$ \end_inset , becomes outdated at time \begin_inset Formula $t_{0}+T$ \end_inset , and exits the cache at time \begin_inset Formula $t_{0}+\tau$ \end_inset . The time \begin_inset Formula $T$ \end_inset until the data item becomes outdated is an Exponential \begin_inset Formula $\left(\mu\right)$ \end_inset random variable such that \begin_inset Formula \begin{align} T & \sim\text{Exp}\left(\mu\right)\\ p\left(T,\mu\right) & =\mu e^{-\mu T}\label{eq:exponential_pdf-1}\\ P\left[T0$ \end_inset , so that the optimal TTL, \begin_inset Formula $\hat{\tau}$ \end_inset is given by \begin_inset Formula \begin{align} \hat{\tau} & =\frac{1}{\lambda_{U}}\left(-1+\sqrt{1+\frac{2\lambda_{U}}{\mu\gamma}}\right)\label{optimal_tau} \end{align} \end_inset Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:optimal_tau" plural "false" caps "false" noprefix "false" \end_inset shows \begin_inset Formula $C$ \end_inset for varying parameters; our small \begin_inset Formula $\mu$ \end_inset approximation appears to provide a reasonable estimate for the optimal value of \begin_inset Formula $\tau$ \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/fig_cost_for_varying_tau.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout The expected cost per unit time \begin_inset Formula $C$ \end_inset of serving a data item for varying \begin_inset Formula $\tau$ \end_inset . The default hidden parameters are \begin_inset Formula $\left(\lambda,\mu,\gamma\right)=\left(10,0.01,100\right)$ \end_inset . The thin vertical lines indicate the optimal TTL \begin_inset Formula $\hat{\tau}$ \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:optimal_tau" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard By applying our estimated item usage rate from Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambda_hat" plural "false" caps "false" noprefix "false" \end_inset ), and estimating the rate at which the data item changes using server logs, Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "optimal_tau" plural "false" caps "false" noprefix "false" \end_inset ) can provide an optimal choice of \begin_inset Formula $\tau$ \end_inset for a single data item, and a single endpoint. However, all of this rests on the assumption that data items are used according to a Poisson process with constant (homogeneous) rate for a period of time \begin_inset Formula $T\gg\tau$ \end_inset . In the next section, we consider whether this is accurate, and consider some dynamical behaviour that arises from modelling item usage as a non-homogen eous Poisson process. \end_layout \begin_layout Section Symbolic dynamics of TTL caching with piecewise-constant periodic rates of data usage \begin_inset CommandInset label LatexCommand label name "sec:Model-for-periodically" \end_inset \end_layout \begin_layout Subsection Non-homogeneous Poisson processes \end_layout \begin_layout Standard So far, we have assumed that data item usage follows a Poisson process with constant rate; this is also referred to as a homogeneous Poisson process. As noted earlier in this chapter, that assumption has been widely used because it is very simple and often adequate for the purpose of constructing a model of a natural counting process. However, it has long been recognised that there are natural counting processes which deviate significantly from the Poisson process. A famous early example of this Ugo Fano's work on the ionization yield of radiation in the 1940s \begin_inset CommandInset citation LatexCommand cite key "fano1946theory,fano1947ionization" literal "false" \end_inset . Fano found that the variation in the number of ionizations in a gas was three times less than what would be predicted by a Poisson process with the same rate of ionization. This work led to the \emph on Fano factor \emph default \begin_inset Formula $F\left(t\right)$ \end_inset \begin_inset CommandInset citation LatexCommand cite key "cox1962renewal" literal "false" \end_inset , defined for a counting process \begin_inset Formula $N\left(t\right)$ \end_inset as \begin_inset Formula \begin{equation} F\left(t\right)=\frac{\left\langle N\left(t\right)-\left\langle N\left(t\right)\right\rangle ^{2}\right\rangle }{\left\langle N\left(t\right)\right\rangle }=\frac{V\left[N\left(t\right)\right]}{E\left[N\left(t\right)\right]} \end{equation} \end_inset or the variance of the process divided by its mean. A variation of the Fano factor is the \emph on index of dispersion \emph default \begin_inset Formula $D$ \end_inset \emph on \emph default \begin_inset CommandInset citation LatexCommand cite key "cox1966statistical" literal "false" \end_inset which takes the form of the Fano factor in the limit \begin_inset Formula $t\rightarrow\infty$ \end_inset \begin_inset Formula \begin{equation} D=\lim_{t\rightarrow\infty}\frac{\left\langle N\left(t\right)-\left\langle N\left(t\right)\right\rangle ^{2}\right\rangle }{\left\langle N\left(t\right)\right\rangle }=\lim_{t\rightarrow\infty}\frac{V\left[N\left(t\right)\right]}{E\left[N\left(t\right)\right]} \end{equation} \end_inset For example, \begin_inset Formula $N_{U}\left(t\right)$ \end_inset has index of dispersion \begin_inset Formula \begin{equation} D_{U}=\frac{\lambda_{U}}{\lambda_{U}}=1 \end{equation} \end_inset which is characteristic of a standard Poisson process. \begin_inset Formula $N_{Q}\left(t\right)$ \end_inset on the other hand, has mean and variance given by \begin_inset Formula \begin{align} \lim_{t\rightarrow\infty}\frac{\left\langle N\left(t\right)\right\rangle }{t} & =\frac{1}{\frac{1}{\lambda_{U}}+\tau}\\ \lim_{t\rightarrow\infty}\frac{\left\langle N\left(t\right)-\left\langle N\left(t\right)\right\rangle ^{2}\right\rangle }{t} & \simeq\frac{1}{\lambda_{U}^{2}\left(\frac{1}{\lambda_{U}}+\tau\right)^{3}} \end{align} \end_inset (see Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Variance-of-query" plural "false" caps "false" noprefix "false" \end_inset ) Hence, \begin_inset Formula $N_{Q}\left(t\right)$ \end_inset has index of dispersion \begin_inset Formula \begin{equation} D_{Q}=\frac{\frac{1}{\lambda_{U}^{2}\left(\frac{1}{\lambda_{U}}+\tau\right)^{3}}}{\frac{1}{\frac{1}{\lambda_{U}}+\tau}}=\frac{1}{\lambda_{U}^{2}\left(\frac{1}{\lambda_{U}}+\tau\right)^{2}}=\frac{1}{\left(1+\lambda_{U}\tau\right)^{2}} \end{equation} \end_inset As \begin_inset Formula $\lambda_{U}$ \end_inset increases, the mean and variance of the stochastic component of the inter-query time decrease. As \begin_inset Formula $\tau$ \end_inset increases, the deterministic component of the inter-query time increases. As \begin_inset Formula $\lambda_{U}\tau\rightarrow\infty$ \end_inset , \begin_inset Formula $D_{Q}\rightarrow0$ \end_inset , and the query process becomes deterministic. \end_layout \begin_layout Standard Once we move beyond the strict definition of the homogeneous Poisson process, there is a wide range of more complex stochastic processes which have been found to be of use in modelling diverse phenomena such as earthquakes \begin_inset CommandInset citation LatexCommand cite key "corral2004long" literal "false" \end_inset , neuron spiking \begin_inset CommandInset citation LatexCommand cite key "maimon2009beyond" literal "false" \end_inset , and spreading dynamics in social networks \begin_inset CommandInset citation LatexCommand cite key "jo2014analytically" literal "false" \end_inset . Some real-world processes are found to be \begin_inset Quotes bld \end_inset bursty', with autocorrelation between event times \begin_inset CommandInset citation LatexCommand cite key "barabasi2005origin,livny1993impact,gusella1991characterizing" literal "false" \end_inset , with particularly relevant examples appearing in the context of Internet communications \begin_inset CommandInset citation LatexCommand cite key "zhao2015seismic,paxson1995wide,leland1994self" literal "false" \end_inset and data networks \begin_inset CommandInset citation LatexCommand cite key "hiraoka2020modeling,d2006data,ye2005stability,sun2010burst" literal "false" \end_inset . Others are modelled using non-homogeneous variations of the Poisson process that have a rate which varies in time \begin_inset CommandInset citation LatexCommand cite key "klein1984time,dimitrov2004periodic,ihler2006adaptive,lawrence2017nonhomogeneous" literal "false" \end_inset . \end_layout \begin_layout Standard To motivate this point in our own context of cybersecurity, we consider query data from a data network in the cybersecurity industry. In this context, endpoints send queries about objects such as files, URLs, and IP addresses, and data items are the current reputations of the queried object based on current information about that object. Our dataset consists of queries made by endpoints in the Republic of Ireland using the McAfee product WSS-LAM, which provides reputations for files. We restrict our attention to a class of files with caching period \begin_inset Formula $\tau=30$ \end_inset minutes over a four week period between midnight on Monday 1st August and midnight on 29th August 2022. The data query data is sampled at a rate \begin_inset Formula $\pi=0.5$ \end_inset , so that every query has a 50% chance of being included in the dataset. In Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:variable_query_rate" plural "false" caps "false" noprefix "false" \end_inset , we show the recorded query volume per hour over 28 days plotted in blue. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/periodic_variable_query_rate.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Total query volume per hour from endpoints using McAfee cybersecurity software in the Republic of Ireland plotted in blue. The average total query volume for daytime on weekdays, average total query volume for daytime on weekends, and average total query volume for nighttime are plotted in red. \begin_inset CommandInset label LatexCommand label name "fig:variable_query_rate" \end_inset \end_layout \end_inset \end_layout \end_inset The query volume varies significantly over time, and is prone to large spikes. However, we can note that total query volume per hour is generally lower in daytime on weekends than in daytime during the week. It is also significantly lower during the night than during the day. The average total query volume for weekday days, weekend days, and nights are plotted in red. The day/night cycle was chosen such that nighttime is the eight hour period starting on a whole hour such that the difference in query volume between night and day is maximised. Even using very simple metrics, we see that in aggregate, the query volume, and hence, the rate of querying, varies periodically and significantly between day and night. This does not exclude the possibility that the process is also a bursty process; that will require further study at the level of individual endpoints. For now, we would like to consider the dynamical effects of periodic piecewise- constant query rate. \end_layout \begin_layout Subsection Stochastic model \end_layout \begin_layout Standard Having seen that query rates vary periodically over time, we now consider a variation of our model where \begin_inset Formula $\pi=1$ \end_inset and \begin_inset Formula $\lambda_{U}$ \end_inset varies according to periodic step function, so that item usage is a piecewise-c onstant Poisson process \begin_inset CommandInset citation LatexCommand cite key "rajaram2005poisson,kim2014call,de2022determining" literal "false" \end_inset . Let \begin_inset Formula \begin{equation} \lambda_{U}\left(t\right)=\begin{cases} \lambda_{1} & \text{if }t\mod1\leq\rho\\ \lambda_{2} & \text{if }t\mod1>\rho \end{cases} \end{equation} \end_inset where \begin_inset Formula $\rho$ \end_inset determines the proportion of the time \begin_inset Formula $\lambda_{U}=\lambda_{1}$ \end_inset . \begin_inset Formula $\lambda_{U}\left(t\right)$ \end_inset is periodic with period 1. If \begin_inset Formula $\lambda_{1}=\lambda_{2}$ \end_inset , the rate is constant, so we assume \begin_inset Formula $\lambda_{1}>\lambda_{2}$ \end_inset . The system now has parameters \begin_inset Formula $\lambda_{1}$ \end_inset , \begin_inset Formula $\lambda_{2}$ \end_inset , \begin_inset Formula $\rho$ \end_inset and \begin_inset Formula $\tau$ \end_inset . With the introduction of the step function, the dynamics of the system now take place on a circle with circumference \begin_inset Formula $1$ \end_inset , and so we observe the dynamics with respect to \begin_inset Formula $\theta=t\mod1$ \end_inset . Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Two-comparable-caching" plural "false" caps "false" noprefix "false" \end_inset shows the result of simulating the system for varying \begin_inset Formula $\rho$ \end_inset and \begin_inset Formula $\tau$ \end_inset with \begin_inset Formula $\lambda_{1}=100$ \end_inset and \begin_inset Formula $\lambda_{2}=0$ \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/fig_mixed_rate_sample_solutions.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Two comparable caching time trajectories for \begin_inset Formula $\lambda_{1}=100$ \end_inset , \begin_inset Formula $\lambda_{2}=0$ \end_inset and \begin_inset Formula $\rho=0.8$ \end_inset . \begin_inset Formula $\tau=0.1$ \end_inset in (a) and \begin_inset Formula $\tau=0.7$ \end_inset in (b). (c) shows the distribution of queries for (a) analytically derived from the Gamma \begin_inset Formula $\left(k,\lambda\right)$ \end_inset distribution. \begin_inset CommandInset label LatexCommand label name "fig:Two-comparable-caching" \end_inset \end_layout \end_inset \end_layout \end_inset Both plots show four peaks where queries take place, but the overall shape is different. Consider Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Two-comparable-caching" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(a\right)$ \end_inset , where \begin_inset Formula $\tau=0.2$ \end_inset . The system starts at \begin_inset Formula $\theta=0$ \end_inset with nothing cached. The time until the data item is queried is an Exp \begin_inset Formula $\left(\lambda_{1}\right)$ \end_inset random variable; hence, the shape of the first peak is given by an Exp \begin_inset Formula $\left(\lambda_{1}\right)$ \end_inset probability density function which starts at \begin_inset Formula $\theta=0$ \end_inset . The time interval between the first and second query is \begin_inset Formula $\tau+$ \end_inset Exp \begin_inset Formula $\left(\lambda_{1}\right)$ \end_inset , so that the second query occurs at time \begin_inset Formula $\text{Exp}\left(\lambda_{1}\right)+\tau+\text{Exp}\left(\lambda_{1}\right)$ \end_inset ,where the random variables are independent. The sum of \begin_inset Formula $k$ \end_inset independent Exp \begin_inset Formula $\left(\lambda\right)$ \end_inset random variables is a Gamma \begin_inset Formula $\left(k,\lambda^{-1}\right)$ \end_inset random variable \begin_inset CommandInset citation LatexCommand cite key "bickel2001mathematical,pawitan2001all" literal "false" \end_inset (see Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Sum-of-Exponential" plural "false" caps "false" noprefix "false" \end_inset ), with probability density function \begin_inset Formula \begin{equation} f\left(x,k,\lambda\right)=\frac{\lambda^{k}x^{k-1}e^{-\lambda x}}{\Gamma\left(k\right)} \end{equation} \end_inset Hence, the second peak has the shape of a Gamma \begin_inset Formula $\left(2,\lambda^{-1}\right)$ \end_inset probability density function, the third has Gamma \begin_inset Formula $\left(3,\lambda^{-1}\right)$ \end_inset shape, and the fourth has a Gamma \begin_inset Formula $\left(4,\lambda^{-1}\right)$ \end_inset shape. There is no fifth peak, as the data item leaves the cache at \begin_inset Formula $\theta>\rho$ \end_inset ; instead, the system arrives at \begin_inset Formula $\theta=0$ \end_inset with nothing in the cache to repeat the cycle. This creates an orbit on the circle where each successive peak is closer to \begin_inset Formula $\left[\rho,1\right]$ \end_inset , eventually entering the interval where \begin_inset Formula $\lambda=0$ \end_inset and returning to \begin_inset Formula $\theta=0$ \end_inset . Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Two-comparable-caching" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(c\right)$ \end_inset shows an analytic reconstruction of the query distribution in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Two-comparable-caching" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(a\right)$ \end_inset using the Gamma \begin_inset Formula $\left(k,\lambda^{-1}\right)$ \end_inset probability density function, which is identical to that found as a histogram. \end_layout \begin_layout Standard If we compare this to Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Two-comparable-caching" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(b\right)$ \end_inset , we note that the shapes of the peak are the same, but out of order. This is due to \begin_inset Formula $\tau$ \end_inset now being large enough that it is possible for the orbit jumps over \begin_inset Formula $\left[\rho,1\right]$ \end_inset from an appropriate starting point. We can further explore the dynamics of this model by observing that the two orbits have a different maximum value for \begin_inset Formula $\theta$ \end_inset . In Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:rho-tau-plane" plural "false" caps "false" noprefix "false" \end_inset , we plot the maximum \begin_inset Formula $\theta$ \end_inset achieved by iterating the model for two different values of \begin_inset Formula $\lambda$ \end_inset . We note that as we increase \begin_inset Formula $\lambda$ \end_inset , a pattern of triangles begins to emerge. To better understand this pattern, we note that as \begin_inset Formula $\lambda\rightarrow\infty$ \end_inset , Exp \begin_inset Formula $\left(\lambda\right)\rightarrow0$ \end_inset , so that the next query happens as soon as the data item leaves the cache. Hence, in the limit \begin_inset Formula $\lambda_{1}\rightarrow\infty$ \end_inset and \begin_inset Formula $\lambda_{2}=0$ \end_inset , the system simplifies to a deterministic map. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/ttl_vs_sq_combined.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Colour maps showing the maximum \begin_inset Formula $\theta$ \end_inset after iterating for varying parameters \begin_inset Formula $\left(\rho,\tau\right)$ \end_inset . \begin_inset Formula $\max\left(\theta\right)$ \end_inset ranges from \begin_inset Formula $0$ \end_inset (dark blue) to 1 (dark red). \begin_inset Formula $\lambda_{1}=50$ \end_inset in the first colour map, and \begin_inset Formula $\lambda_{2}=100$ \end_inset in the second. A distinctive pattern emerges as \begin_inset Formula $\lambda_{1}$ \end_inset increases. \begin_inset CommandInset label LatexCommand label name "fig:rho-tau-plane" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Deterministic limit as a circle map \end_layout \begin_layout Standard We construct a map \begin_inset Formula \begin{equation} \theta_{n+1}=f\left(\theta_{n}\right) \end{equation} \end_inset where if the item is queried at time \begin_inset Formula $\theta_{n}$ \end_inset , \begin_inset Formula $\theta_{n+1}$ \end_inset is the next time at which the item is queried, such that \begin_inset Formula \begin{equation} f\left(\theta;\rho,\tau\right)=\begin{cases} \theta+\tau & \text{if }\theta+\tau<\rho\\ 0 & \text{if }\rho\leq\theta+\tau<1\\ \left(\theta+\tau\right)\mod1 & \text{if }1\leq\theta+\tau \end{cases}\label{eq:the_map} \end{equation} \end_inset as shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:analytic_map" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/fig_analytic_map.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Plot of the analytic map \begin_inset Formula $f\left(\theta\right)$ \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:analytic_map" \end_inset \end_layout \end_inset \end_layout \end_inset This map is comparable to the circle map \begin_inset CommandInset citation LatexCommand cite key "ott2002chaos" literal "false" \end_inset ; we can write Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:the_map" plural "false" caps "false" noprefix "false" \end_inset ) in the form \begin_inset Formula \begin{equation} \theta_{n+1}=\tau+g\left(\theta\right) \end{equation} \end_inset where \begin_inset Formula $g\left(t\right)$ \end_inset is a nonlinear function \begin_inset Formula \begin{equation} g\left(\theta\right)=\begin{cases} 0 & \theta+\tau<\rho\\ 1-\left(\theta+\tau\right) & \theta+\tau\geq\rho \end{cases} \end{equation} \end_inset The circle map has been studied for a wide variety of nonlinear functions \begin_inset Formula $g$ \end_inset \begin_inset CommandInset citation LatexCommand cite key "ott2002chaos,arnold1991cardiac,glass1991cardiac,glass2001synchronization,jensen1983complete,mcguinness2004arnold,briggs1999anatomy" literal "false" \end_inset , including a number of discontinuous functions \begin_inset CommandInset citation LatexCommand cite key "derks2021creation,coombes1996neuronal,bauer1992new,qu1997multiple,bailey2018circle,granados2017period,lajoie2011shared" literal "false" \end_inset . The unique feature of our map is that it is linear within \begin_inset Formula $\left[0,\rho\right]$ \end_inset , but contracts \begin_inset Formula $\left[\rho,1\right]$ \end_inset to a point. We iterate the map for varying \begin_inset Formula $\left(\rho,\tau\right)$ \end_inset , which can be seen in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Deterministic-limit" plural "false" caps "false" noprefix "false" \end_inset . We observe that the distinctive pattern of triangles seen in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Two-comparable-caching" plural "false" caps "false" noprefix "false" \end_inset have resolved clearly. As \begin_inset Formula $\tau$ \end_inset increases, \begin_inset Formula $\max\left(\theta\right)$ \end_inset increases linearly within each triangle. However, at the boundary of each triangle, \begin_inset Formula $\max\left(\theta\right)$ \end_inset changes discontinuously. To understand why, we consider the system through a symbolic representation. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/fig_analytic_model_max.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Maximum observed value of \begin_inset Formula $\theta$ \end_inset for orbits of the unit circle under the map \begin_inset Formula $\theta_{n+1}=f\left(\theta_{n}\right)$ \end_inset which begin at \begin_inset Formula $\theta_{0}=0$ \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:Deterministic-limit" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Symbolic dynamics \end_layout \begin_layout Standard We recall the interesting ordering effect in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:Two-comparable-caching" plural "false" caps "false" noprefix "false" \end_inset , where the shape of a peak was determined by its order in the orbit. Similarly, \begin_inset Formula $f\left(\theta;\tau,\rho\right)$ \end_inset generates an orbit \begin_inset Formula $\left\{ f^{k}\left(\theta;\tau,\rho\right)\right\} _{k=0}^{\infty}$ \end_inset which can be represented symbolically in terms of where a particular point maps to; left of \begin_inset Formula $\rho$ \end_inset , between \begin_inset Formula $\rho$ \end_inset and \begin_inset Formula $1$ \end_inset , or right of \begin_inset Formula $1$ \end_inset . We can define this by \begin_inset Formula \begin{equation} s\left(\theta;\rho,\tau\right)=\begin{cases} L & \text{if }\theta+\tau<\rho\\ C & \text{if }\rho\leq\theta+\tau<1\\ R & \text{if }1\leq\theta+\tau \end{cases} \end{equation} \end_inset A periodic orbit can be defined by a periodic sub-sequence of symbols of length \begin_inset Formula $P$ \end_inset . In Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic_colour_map" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset Formula $P$ \end_inset is plotted for varying parameters. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/fig_analytic_model_count.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Period \begin_inset Formula $P$ \end_inset for orbits of the unit circle under the map \begin_inset Formula $\theta_{n+1}=f\left(\theta_{n}\right)$ \end_inset which begin at \begin_inset Formula $\theta_{0}=0$ \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:symbolic_colour_map" \end_inset \end_layout \end_inset \end_layout \end_inset We observe that the pattern of triangles observed before is more clear, and conclude that the pattern is associated with changing sequences. Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic_labelled" plural "false" caps "false" noprefix "false" \end_inset shows the sequences for some of the larger tongues. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/fig_analytic_model_count_gs_labelled.pdf width 85text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Period \begin_inset Formula $P$ \end_inset for orbits of the unit circle under the map \begin_inset Formula $\theta_{n+1}=f\left(\theta_{n}\right)$ \end_inset which begin at \begin_inset Formula $\theta_{0}=0$ \end_inset (identical to Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic_colour_map" plural "false" caps "false" noprefix "false" \end_inset ). Some of the larger tongues have been labelled with the sequence of their orbit. \begin_inset CommandInset label LatexCommand label name "fig:symbolic_labelled" \end_inset \end_layout \end_inset \end_layout \end_inset There are multiple tongues of the same period, but each tongue has a unique sequence representation in terms of the symbols \begin_inset Formula $L$ \end_inset , \begin_inset Formula $R$ \end_inset , and \begin_inset Formula $C$ \end_inset . However, a period \begin_inset Formula $n+1$ \end_inset sequence has \begin_inset Formula $n+1$ \end_inset symbols, which allows for a total of \begin_inset Formula $2^{n}$ \end_inset sequence permutations (the \begin_inset Formula $C$ \end_inset at the end is constant). There are only two observed period 6 orbits, but \begin_inset Formula $2^{6}=32$ \end_inset possible sequences. For sequences of length \begin_inset Formula $n+1$ \end_inset , we observe up to \begin_inset Formula $n$ \end_inset different tongues; this is only observed when \begin_inset Formula $n+1$ \end_inset is prime. This means that for a sequence of a given length, only a small fraction of them are \emph on legal \emph default , representing a orbit that can exist. Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Sample-orbits" plural "false" caps "false" noprefix "false" \end_inset shows all legal sequences which end at \begin_inset Formula $C$ \end_inset up to period 5 with orbits for given \begin_inset Formula $\left(\rho,\tau\right)$ \end_inset . \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \noindent \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(\rho,\tau\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Sequence \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Orbit in \begin_inset Formula $\theta$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(0.33,0.67\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $C$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left\{ 0\right\} $ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(0.50,0.33\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $LC$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left\{ 0.33,0\right\} $ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(0.56,0.22\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $LLC$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left\{ 0.22,0.44,0\right\} $ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(0.72,0.61\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $LRC$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left\{ 0.61,0.22,0\right\} $ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(0.58,0.17\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $LLLC$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left\{ 0.17,0.34,0.51,0\right\} $ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(0.81,0.72\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $LRRC$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left\{ 0.72,0.44,0.16,0\right\} $ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(0.60,0.13\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $LLLLC$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left\{ 0.13,0.26,0.39,0.52,0\right\} $ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(0.85,0.78\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $LRRRC$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left\{ 0.78,0.56,0.34,0.12,0\right\} $ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(0.77,0.57\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $LRLRC$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left\{ 0.57,0.14,0.71,0.28,0\right\} $ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left(0.82,0.38\right)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $LLRLC$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\left\{ 0.38,0.76,0.14,0.52,0\right\} $ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Sample orbits for unique legal sequences up to period 5 which end at \begin_inset Formula $C$ \end_inset . \begin_inset CommandInset label LatexCommand label name "tab:Sample-orbits" \end_inset \end_layout \end_inset \end_layout \end_inset Given a sequence, we can generate a set of inequalities using \begin_inset Formula $s\left(\theta;\rho,\tau\right)$ \end_inset , which indicate a region of the plane where the orbit exists. For example, the sequence \begin_inset Formula $LRC$ \end_inset yields inequalities \begin_inset Formula \begin{align} f\left(\theta\right) & =\tau\mod1<\rho\\ f^{2}\left(\theta\right) & =\left(2\tau\right)\mod1\geq1\\ f^{3}\left(\theta\right) & =\left(3\tau\right)\mod1<1\\ f^{3}\left(\theta\right) & =\left(3\tau\right)\mod1\geq\rho \end{align} \end_inset However, the low ratio of legal sequences to illegal sequences makes this an inefficient method of searching for legal sequences. Fortunately, we can determine rules for how legal sequences are generated based on observations from Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic_labelled" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Enumerate For \begin_inset Formula $\tau\geq\rho$ \end_inset , the only orbit is the \begin_inset Formula $C$ \end_inset orbit. This occurs because for \begin_inset Formula $\tau\geq\rho$ \end_inset , any orbit that starts at \begin_inset Formula $\theta=0$ \end_inset will map to the \begin_inset Formula $C$ \end_inset region, and get sent back to 0. \end_layout \begin_layout Enumerate For \begin_inset Formula $\tau<1-\rho$ \end_inset , there are no orbits containing \begin_inset Formula $R$ \end_inset symbols, as there is no \begin_inset Formula $\theta<\rho$ \end_inset such that \begin_inset Formula $\theta+\tau\geq1$ \end_inset . \end_layout \begin_layout Enumerate All tongues without an \begin_inset Formula $R$ \end_inset in their name connect to the origin and follow a common rule. As \begin_inset Formula $\tau$ \end_inset decreases from \begin_inset Formula $\tau=\rho$ \end_inset , the integer \begin_inset Formula $k$ \end_inset such that \begin_inset Formula $k\tau<\rho\leq\left(k+1\right)\tau$ \end_inset increases, and hence, the period of the orbit increases. This generates a set of tongues fanning out from the origin. Starting from the \begin_inset Formula $C$ \end_inset tongue and moving clockwise, add an \begin_inset Formula $L$ \end_inset in front of the \begin_inset Formula $C$ \end_inset to find the sequence for the next tongue, generating \begin_inset Formula $C$ \end_inset , \begin_inset Formula $LC$ \end_inset , \begin_inset Formula $LLC$ \end_inset , \begin_inset Formula $LLLC$ \end_inset , ... \begin_inset Formula $L^{k}C$ \end_inset . \end_layout \begin_layout Enumerate Tongues with an \begin_inset Formula $R$ \end_inset in their sequence are born from adjacent lower period tongues. A higher period \begin_inset Quotes bld \end_inset child' tongue has two adjacent lower period \begin_inset Quotes bld \end_inset parent' tongues. The left parent meets the child along its left edge. The lower parent meets the child tongue at its lower left vertex. In order to derive the sequence for the child tongue, start with the sequence of the lower parent. Change the \begin_inset Formula $C$ \end_inset at the end of the sequence to an \begin_inset Formula $R$ \end_inset , then append the sequence of the left parent. For example, the tongue containing the sequence \begin_inset Formula $LRC$ \end_inset has \begin_inset Formula $C$ \end_inset as left parent and \begin_inset Formula $LC$ \end_inset as lower parent. \end_layout \begin_layout Standard Using these rules, we can find orbits that exist, and then use \begin_inset Formula $s\left(\theta;\rho,\tau\right)$ \end_inset to find regions of the \begin_inset Formula $\left(\rho,\tau\right)$ \end_inset plane where they exist. \end_layout \begin_layout Paragraph Theorem: \end_layout \begin_layout Standard If \begin_inset Formula $\tau,\rho\in\left(0,1\right)$ \end_inset then every orbit that begins at \begin_inset Formula $\theta=0$ \end_inset must pass through \begin_inset Formula $\theta=0$ \end_inset again. \end_layout \begin_layout Paragraph Proof: \end_layout \begin_layout Standard Consider the circle map \begin_inset Formula \begin{equation} h\left(\theta;\tau\right)=\left(\theta+\tau\right)\mod1 \end{equation} \end_inset and note that \begin_inset Formula \begin{equation} h\left(\theta;\tau\right)=f\left(\theta;\tau,1\right) \end{equation} \end_inset \begin_inset Formula $h\left(\theta;\tau\right)$ \end_inset defines an orbit \begin_inset Formula $\left\{ h^{k}\left(\theta;\tau\right)\right\} _{k=1}^{\infty}$ \end_inset on the unit circle. An \begin_inset Formula $n$ \end_inset -periodic orbit is an orbit such that such that \begin_inset Formula \begin{equation} h^{n}\left(\theta;\tau\right)=\theta \end{equation} \end_inset Then an orbit beginning at \begin_inset Formula $\theta=0$ \end_inset is defined by \begin_inset Formula \begin{align} h^{n}\left(0\right) & =\left(n\tau\right)\mod1 \end{align} \end_inset From here, we consider two cases. \end_layout \begin_layout Paragraph \begin_inset Formula $\left(a\right)$ \end_inset \end_layout \begin_layout Standard First, consider rational \begin_inset Formula $\tau\in\mathbb{Q}$ \end_inset . If \begin_inset Formula $\tau$ \end_inset is rational, then \begin_inset Formula \begin{equation} \tau=\frac{m}{n} \end{equation} \end_inset for some integers \begin_inset Formula $m,n\in\mathbb{N}$ \end_inset . Then \begin_inset Formula \begin{align} h^{n}\left(0\right) & =\left(n\frac{m}{n}\right)\mod1\\ & =\left(m\right)\mod1\\ & =0 \end{align} \end_inset Hence, \begin_inset Formula $h^{n}\left(0\right)$ \end_inset is \begin_inset Formula $n$ \end_inset -periodic. Let \begin_inset Formula \begin{equation} n_{f}=\begin{cases} n & \text{if \ensuremath{g^{k}\left(0\right)<\rho} for \ensuremath{k=1,2,...,n}}\\ \min\left\{ k:\rho\leq h^{k}\left(0\right)<1\right\} & \text{o.w.} \end{cases} \end{equation} \end_inset Then the orbit \begin_inset Formula $\left\{ f^{k}\left(\theta;\tau,\rho\right)\right\} _{k=1}^{\infty}$ \end_inset is \begin_inset Formula $n_{f}$ \end_inset -periodic. \begin_inset Formula $\left\{ f^{k}\left(\theta;\tau,\rho\right)\right\} _{k=1}^{\infty}$ \end_inset is at most \begin_inset Formula $n$ \end_inset -periodic, but the orbit can return to \begin_inset Formula $0$ \end_inset earlier if a point on the orbit enters \begin_inset Formula $\left[\rho,1\right]$ \end_inset . \end_layout \begin_layout Paragraph \begin_inset Formula $\left(b\right)$ \end_inset \end_layout \begin_layout Standard Next, consider irrational \begin_inset Formula $\tau\in\mathbb{R}\backslash\mathbb{Q}$ \end_inset . Then \begin_inset Formula $\left\{ h^{k}\left(\theta;\tau\right)\right\} _{k=1}^{\infty}$ \end_inset is dense in \begin_inset Formula $\left[0,1\right]$ \end_inset \begin_inset CommandInset citation LatexCommand cite key "kuipers2012uniform,ott2002chaos" literal "false" \end_inset , and therefore the orbit passes arbitrarily close to every point in \begin_inset Formula $\left[\rho,1\right]$ \end_inset . Hence, there are points in \begin_inset Formula $\left\{ h^{k}\left(\theta;\tau\right)\right\} _{k=1}^{\infty}$ \end_inset which enter \begin_inset Formula $\left[\rho,1\right]$ \end_inset , and so \begin_inset Formula $\left\{ f^{k}\left(\theta;\tau,\rho\right)\right\} _{k=1}^{\infty}$ \end_inset must also include points in \begin_inset Formula $\left[\rho,1\right],$ \end_inset which map to \begin_inset Formula $\theta=0$ \end_inset . \end_layout \begin_layout Standard Therefore, if \begin_inset Formula $\tau,\rho\in\left(0,1\right)$ \end_inset , then every orbit that begins at \begin_inset Formula $\theta=0$ \end_inset must pass through \begin_inset Formula $\theta=0$ \end_inset again. \begin_inset Formula $\blacksquare$ \end_inset \end_layout \begin_layout Subsection Orbits that do not pass through \begin_inset Formula $\theta=0$ \end_inset \end_layout \begin_layout Standard Above, we consider orbits which start at \begin_inset Formula $\theta=0$ \end_inset and end at \begin_inset Formula $\theta=0$ \end_inset , which must therefore contain \begin_inset Formula $C$ \end_inset in their sequence. However, these are not the only possible orbits. Consider \begin_inset Formula $\tau=\frac{1}{3}$ \end_inset and \begin_inset Formula $\rho>\frac{5}{6}$ \end_inset . Let \begin_inset Formula $\theta=\frac{1}{2}$ \end_inset , then \begin_inset Formula \begin{equation} \theta=\frac{1}{2},\quad f\left(\theta\right)=\frac{5}{6},\quad f^{2}\left(\theta\right)=\frac{1}{6},\quad f^{3}\left(\theta\right)=\frac{1}{2} \end{equation} \end_inset This yields a 3-periodic orbit with sequence \begin_inset Formula $LRR$ \end_inset which does not contain \begin_inset Formula $C$ \end_inset . Such orbits are only possible for rational \begin_inset Formula $\tau>1-\rho$ \end_inset ; otherwise, the orbit would be dense in \begin_inset Formula $\left[\rho,1\right]$ \end_inset , or wouldn't be able to jump \begin_inset Formula $\left[\rho,1\right]$ \end_inset . Considering initial \begin_inset Formula $\theta>0$ \end_inset extends the original problem into an additional dimension, and so it would be be interesting to study such orbits to see how they compare to orbits that start at \begin_inset Formula $\theta=0$ \end_inset . \end_layout \begin_layout Standard \end_layout \begin_layout Section Conclusion \end_layout \begin_layout Standard In this chapter, we construct a model for generating query data from a data network which features caching to reduce query volume and random sampling to reduce record volume. By modelling data usage as a Poisson process, we estimate the rate of data usage based on the rate of data recording. We expand our model to estimate the optimal caching period for volatile data, and finally, we consider the dynamical effect of non-homogeneous data usage rates through the lens of symbolic dynamics in a deterministic limit of our model. \end_layout \begin_layout Standard As noted at the beginning of this chapter, the material presented requires further development before it wilinvl be suitable for publication. This further development may be undertaken in several different directions. To take the first section as an example, the model of query data generation is somewhat simple due to our choice of a Poisson process for data usage. The Poisson process is attractive because it is simple and relatively tractable. One interesting avenue of enquiry would be to continue to study the presented version of the model and derive analytic results for distributions and moments. However, we have noted that there are applications where the Poisson process is not sufficiently accurate, and that this may be one of those applications. In that case, it would be of practical interest to extend the model to a more general data usage process. Even then, a full analysis of the Poisson version would make a valuable point of reference. \end_layout \begin_layout Standard Our extension to the model for finding the optimal caching period would also be worth considering in the light of a more general process. For example, if data is used in bursts of activity, followed by periods of inactivity, then we would expect that the optimal caching period is correlated with the expected length of a burst. We might also consider expanding the model to multiple endpoints and multiple items. In such a case, assigning a unique caching period to each endpoint/item pair might be unnecessarily complex, and would complicate any analysis of query data generated by the data network. In practice, caching periods are determined by assigning items to different classes; for example, the data shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:variable_query_rate" plural "false" caps "false" noprefix "false" \end_inset consisted of queries for data files which had not yet been positively identifie d as being either safe or unsafe. Such data is likely to change as more information is collected, so the caching period is short. As confidence in identification increases, so can the caching period. Identifying malicious files and URLs is an evolving battle between cybersecurit y providers and malware creators, but the manner in which that information is used remains the same. Hence, further analysis of how to optimise the propagation of this information will always be of value. \end_layout \begin_layout Standard Finally, our circle map model generates interesting figures which describe a hierarchical pattern that can be understand through symbolic dynamics. This model is a significant abstraction from the original stochastic model, and so it may lack the same level of applicability as the other sections within the context of cybersecurity. However, models like this can help us identify signals that are otherwise lost in the noise of of the query data. Outside of this context, the circle map is so ubiquitous that every opportunity to understand it is worth considering. As noted in the final subsection, there is a clear path to further research here by considering orbits that do not pass through the point \begin_inset Formula $\theta=0$ \end_inset , and how those orbits connect with the ones studied in this section. We also note that the striking hierarchical pattern of triangles visible in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:symbolic_colour_map" plural "false" caps "false" noprefix "false" \end_inset may feature some degree of self-similarity, and it would be worth investigating whether a transformation can be found which would map the pattern to some subset of itself. \end_layout \begin_layout Standard There is clearly much scope for further work on this topic, falling into a number of areas of mathematics, and as many internet systems rely on efficient and reliable dissemination of data, we expect that this work will be interest to a wide audience. \end_layout \begin_layout Chapter* Conclusion & Outlook \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash phantomsection \end_layout \begin_layout Plain Layout \backslash addcontentsline{toc}{chapter}{Conclusion} \end_layout \begin_layout Plain Layout \backslash renewcommand{ \backslash leftmark}{Conclusion} \end_layout \end_inset The Internet is a highly complex system that has a significant impact on our world. It is, therefore, crucial that we understand its dynamical behaviour. In this thesis, we formalise small-scale problems derived from Internet systems mathematically so that their dynamics can be studied rigorously. We observe that stochasticity is a crucial feature in modelling the negotiation s following a targeted ransomware attack, while modelling the server/endpoint interactions of a data network requires dynamical features such as time delay, periodic forcing, switching and stochasticity. In addition, we study two dynamical systems from climate science and signal processing which feature time delay, switching, and either periodic forcing or linear flow. These systems give rise to complex resonance phenomena and rich bifurcation structures. \end_layout \begin_layout Standard In Chapter 1, we studied a dynamical system derived from a climate science model which featured switched time-delayed feedback and switched periodic forcing. We observed a rich structure of torus bifurcations spanned by Arnold tongues divided into \begin_inset Quotes bld \end_inset strings of sausages' by zero-width shrinking points. Solutions within the same Arnold tongue had the same period \begin_inset Formula $P$ \end_inset , but the number of zero crossings \begin_inset Formula $R$ \end_inset per period varied along the string. Due to the switching, we were able to study the dynamics of this non-smooth system analytically using a symbolic representation. This enabled us to map out key features of the bifurcation structure analytical ly, something which could only be achieved numerically in earlier smooth systems \begin_inset CommandInset citation LatexCommand citep key "ghil_delay_2008" literal "true" \end_inset . One area of particular interest which could merit further study is the relationship between the Chenciner (generalised Neimark-Sacker) bifurcations and the curve which bounds the region inside which we observe Arnold tongues of stable solutions with characteristic ratios \begin_inset Formula $P:R$ \end_inset where \begin_inset Formula $R0$ \end_inset . \end_layout \begin_layout Itemize Each event occurring between consecutive \begin_inset Formula $\bar{Z}$ \end_inset and \begin_inset Formula $Z$ \end_inset events occurs at \begin_inset Formula $x<0$ \end_inset . \end_layout \begin_layout Standard If the solved system of equations satisfies the set of inequalities then the sequence is legal, for a given \begin_inset Formula $\left(b,\tau\right)$ \end_inset . These techniques allow us to use symbolic sequences to study solutions systematically. Solving for the times and positions associated with events in a sequence, we can plot the solution represented by the sequence. Additionally, by changing the inequalities to equalities, we obtain the bifurcation curves shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:bifurcations" \end_inset . \end_layout \begin_layout Subsection Calculation of the bifurcation curve of \begin_inset Formula $\mathbb{P}$ \end_inset \begin_inset CommandInset label LatexCommand label name "subsec:Calculation-of-the" \end_inset \end_layout \begin_layout Standard For \begin_inset Formula $\tau>\frac{1}{2}$ \end_inset , \begin_inset Formula $\mathbb{P}$ \end_inset can written as \begin_inset Formula \begin{equation} \mathbb{P}\left(S_{z}\right)=AS_{z}+B \end{equation} \end_inset Due to the sparsity of \begin_inset Formula $A$ \end_inset , the characteristic equation can be readily calculated as \begin_inset Formula \begin{equation} (-1-\lambda)(-\lambda)^{n-1}+(-1)^{n-1}\frac{2(-1)^{n-1}}{b+(-1)^{n-1}}=0 \end{equation} \end_inset This simplifies to \begin_inset Formula \begin{equation} \lambda^{n}+\lambda^{n-1}-\frac{2(-1)^{n-1}}{b+(-1)^{n-1}}=0 \end{equation} \end_inset By substituting \begin_inset Formula $\lambda=e^{i\rho}$ \end_inset , we can solve for \begin_inset Formula $b$ \end_inset to determine \begin_inset Formula $b=b_{\text{bif }}$ \end_inset for which the fixed point of \begin_inset Formula $\mathbb{P}$ \end_inset is bifurcating as \begin_inset Formula \begin{equation} e^{i\rho n}+e^{i\rho\left(n-1\right)}-\frac{2(-1)^{n-1}}{b_{\text{bif}}+(-1)^{n-1}}=0\label{eq:a} \end{equation} \end_inset Note that as \begin_inset Formula $\frac{2(-1)^{n-1}}{b_{\text{bif}}+(-1)^{n-1}}\in\mathbb{R}$ \end_inset , \begin_inset Formula \begin{equation} e^{i\rho n}=\overline{e^{i\rho\left(n-1\right)}}\label{eq:b} \end{equation} \end_inset So, \begin_inset Formula $e^{i\rho\left(2n-1\right)}=1=e^{i2\pi k},k\in\mathbb{Z}$ \end_inset , and so \begin_inset Formula $\rho=\frac{2\pi k}{2n-1}$ \end_inset . In order to satisfy Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:b" \end_inset ), \begin_inset Formula $k=n-1$ \end_inset . Substituting \begin_inset Formula $\rho$ \end_inset back into Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:a" \end_inset ), \begin_inset Formula \begin{equation} \cos\left(\frac{2\pi n(n-1)}{2n-1}\right)+\cos\left(\frac{2\pi\left(n-1\right)^{2}}{2n-1}\right)-\frac{2(-1)^{n-1}}{b_{\text{bif}}+(-1)^{n-1}}=0 \end{equation} \end_inset which simplifies under a sum-to-product cosine identity to \begin_inset Formula \begin{equation} \cos\left(\frac{\pi\left(n-1\right)}{2n-1}\right)-\frac{1}{b_{\text{bif}}+(-1)^{n-1}}=0 \end{equation} \end_inset Therefore, \begin_inset Formula \begin{equation} b_{\text{bif}}(n)=\frac{1}{\cos\left(\frac{\pi\left(n-1\right)}{2n-1}\right)}-(-1)^{n-1} \end{equation} \end_inset where \begin_inset Formula $n=\left\lceil 2\tau\right\rceil $ \end_inset . \end_layout \begin_layout Section Dynamics of a band-pass filter system with switched time-delayed feedback \end_layout \begin_layout Subsection Derivation of the bandpass-filter system \begin_inset CommandInset label LatexCommand label name "subsec:Derivation-of-the" \end_inset \end_layout \begin_layout Standard A low-pass filter is a system or device that passes frequencies lower than the corner frequency \begin_inset Formula $\omega_{L}$ \end_inset and attenuates signals with frequency higher than \begin_inset Formula $\omega_{L}$ \end_inset \begin_inset CommandInset citation LatexCommand cite key "udaltsov2002bandpass,blakely2004high,illing2005hopf" literal "false" \end_inset . A high-pass filter is a system or device that passes frequencies higher than the corner frequency \begin_inset Formula $\omega_{H}$ \end_inset and attenuates signals with frequency lower than \begin_inset Formula $\omega_{H}$ \end_inset . The dynamics of low-pass and high-pass filter systems are described in Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lowpass" plural "false" caps "false" noprefix "false" \end_inset ) and Eq.( \begin_inset CommandInset ref LatexCommand ref reference "eq:lowpass" plural "false" caps "false" noprefix "false" \end_inset ) respectively \begin_inset CommandInset citation LatexCommand cite key "udaltsov2002bandpass,illing2005hopf,blakely2004high" literal "false" \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \tau_{L}\dot{x}_{L}+x_{L} & =f\left(t\right)\label{eq:lowpass}\\ \dot{x}_{H}+\frac{x_{H}}{\tau_{H}} & =\frac{d}{dt}\left[g\left(t\right)\right]\label{eq:highpass} \end{align} \end_inset where \begin_inset Formula $\tau_{L}=\omega_{L}^{-1}$ \end_inset and \begin_inset Formula $\tau_{H}=\omega_{H}^{-1}$ \end_inset , and \begin_inset Formula $f$ \end_inset and \begin_inset Formula $g$ \end_inset are the input signals which drive the filters. When undriven, the filters relax to \begin_inset Formula $x_{L}=x_{H}=0$ \end_inset . When we drive Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lowpass" plural "false" caps "false" noprefix "false" \end_inset ) and Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:highpass" plural "false" caps "false" noprefix "false" \end_inset ) with input signal \begin_inset Formula $f\left(t\right)=g\left(t\right)=\cos\left(\omega t\right)$ \end_inset , the outputs \begin_inset Formula $x_{L}\left(t\right)$ \end_inset and \begin_inset Formula $x_{H}\left(t\right)$ \end_inset are signals signal with frequency \begin_inset Formula $\omega$ \end_inset whose amplitude depends on \begin_inset Formula $\omega$ \end_inset relative to the corner frequency, as shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:High-pass-and-low-pass" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula $\left(a\right)$ \end_inset and \begin_inset Formula $\left(b\right)$ \end_inset respectively. \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../lucas_system/fig_lowhighpass_filter.pdf width 75text% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Low-pass and high-pass filter output amplitude with input \begin_inset Formula $f$ \end_inset \begin_inset Formula $\left(t\right)=\cos\left(\omega t\right)$ \end_inset for varying input frequency \begin_inset Formula $\omega$ \end_inset , where the corner frequency \begin_inset Formula $\omega_{L}=\omega_{H}=1$ \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:High-pass-and-low-pass" \end_inset \end_layout \end_inset \end_layout \end_inset A band-pass filter can be constructed by coupling a low-pass filter with a high-pass filter \begin_inset CommandInset citation LatexCommand cite key "udaltsov2002bandpass,blakely2004high,illing2005hopf" literal "false" \end_inset . This is achieved by letting \begin_inset Formula $g\left(t\right)=x_{L}\left(t\right)$ \end_inset , yielding the equations \begin_inset Formula \begin{align} \tau_{L}\dot{x}_{L}+x_{L} & =f\left(t\right)\label{eq:lowpass-2}\\ \dot{x}_{H}+\frac{x_{H}}{\tau_{H}} & =\dot{x}_{L}\label{eq:highpass-2} \end{align} \end_inset The system can be simplified by first integrating Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:highpass-2" plural "false" caps "false" noprefix "false" \end_inset ) to yield \begin_inset Formula \begin{equation} x_{H}+\frac{1}{\tau_{H}}\intop_{0}^{t}x_{H}\left(t'\right)dt'=x_{L}\label{eq:integrated} \end{equation} \end_inset with the integration constant being satisfied by the lower bound of the integral. Substituting Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:highpass-2" plural "false" caps "false" noprefix "false" \end_inset ) and Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:integrated" plural "false" caps "false" noprefix "false" \end_inset ) into Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lowpass-2" plural "false" caps "false" noprefix "false" \end_inset ) yields \begin_inset Formula \[ \tau_{L}\left[\dot{x}_{H}+\frac{x_{H}}{\tau_{H}}\right]+\left[x_{H}+\frac{1}{\tau_{H}}\intop_{0}^{t}x_{H}\left(s\right)ds\right]=f\left(t\right) \] \end_inset which we rewrite as \begin_inset Formula \begin{align*} x_{H}+\frac{\tau_{H}\tau_{L}}{\left(\tau_{H}+\tau_{L}\right)}\dot{x}_{H}+\frac{1}{\left(\tau_{H}+\tau_{L}\right)}\intop_{0}^{t}x_{H}\left(s\right)ds & =\frac{\tau_{H}}{\left(\tau_{H}+\tau_{L}\right)}f\left(t\right) \end{align*} \end_inset where \begin_inset Formula $\frac{\tau_{H}}{\left(\tau_{H}+\tau_{L}\right)}$ \end_inset is the signal gain due to passing through the band-pass filter. We substitute centre frequency \begin_inset Formula $\Omega=\frac{1}{\sqrt{\tau_{H}\tau_{L}}}$ \end_inset and quality factor \begin_inset Formula $Q=\frac{\sqrt{\tau_{H}\tau_{L}}}{\left(\tau_{H}+\tau_{L}\right)}$ \end_inset to yield \begin_inset Formula \begin{align} x_{H}+\frac{Q}{\Omega}\frac{dx_{H}}{d\bar{t}}+Q\Omega\intop_{0}^{t}x_{H}\left(s\right)ds & =\frac{\tau_{H}}{\left(\tau_{H}+\tau_{L}\right)}f\left(t\right)\label{eq:bandpass} \end{align} \end_inset We rescale \begin_inset Formula $x_{H}$ \end_inset to get dimensionless output \begin_inset Formula $x=x_{H}\left(\frac{\tau_{H}}{\left(\tau_{H}+\tau_{L}\right)}\right)^{-1}$ \end_inset , yielding the equation \begin_inset Formula \begin{equation} x+\frac{Q}{\Omega}\frac{dx}{dt}+Q\Omega\intop^{t}x\left(s\right)ds=f\left(t\right) \end{equation} \end_inset Finally, we introduce a variable \begin_inset Formula $y=Q\Omega\intop^{t}x\left(s\right)ds$ \end_inset such that \begin_inset Formula $\frac{dy}{dt}=Q\Omega x$ \end_inset , yielding the non-smooth delay differential equation \begin_inset Formula \begin{equation} \begin{aligned}Q\Omega^{-1}\dot{x} & =-x-y+f\left(t\right)\\ \dot{y} & =Q\Omega x \end{aligned} \label{eq:dde-1} \end{equation} \end_inset previously studied in \begin_inset CommandInset citation LatexCommand cite key "udaltsov2002bandpass,illing2005hopf,blakely2004high" literal "false" \end_inset with different input signals \begin_inset Formula $f$ \end_inset \begin_inset Formula $\left(t\right)$ \end_inset . \end_layout \begin_layout Subsection Useful properties of Pauli matrices \end_layout \begin_layout Standard The Pauli matrices, with the identify matrix \begin_inset Formula $I$ \end_inset , form a basis for the vector space of \begin_inset Formula $2\times2$ \end_inset real matrices \begin_inset CommandInset citation LatexCommand cite key "condon1929quantum" literal "false" \end_inset . They are \begin_inset Formula \begin{align} I & =\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right)\\ \sigma_{x} & =\left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right)\\ \sigma_{y} & =\left(\begin{array}{cc} 0 & -i\\ i & 0 \end{array}\right)\\ \sigma_{z} & =\left(\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right) \end{align} \end_inset Pauli matrices have some very useful properties. Firstly, the square of any Pauli matrix \begin_inset Formula $\sigma_{j}$ \end_inset yields the identity matrix \begin_inset Formula \begin{equation} \left(\sigma_{i}\right)^{2}=I \end{equation} \end_inset Secondly, Pauli matrices anti-commute; for a pair of different Pauli matrices \begin_inset Formula $\sigma_{j}$ \end_inset and \begin_inset Formula $\sigma_{k}$ \end_inset , \begin_inset Formula $j\neq k$ \end_inset , \begin_inset Formula \begin{equation} \sigma_{j}\sigma_{k}=-\sigma_{k}\sigma_{j} \end{equation} \end_inset These two properties can be summarised in Einstein notation \begin_inset CommandInset citation LatexCommand cite key "einstein1916die" literal "false" \end_inset as \begin_inset Formula \begin{equation} \sigma_{j}\sigma_{k}=\delta_{jk}I+i\epsilon_{jkl}\sigma_{l} \end{equation} \end_inset As the Pauli matrices and the identify matrix form a basis for the vector space of \begin_inset Formula $2\times2$ \end_inset real matrices, a \begin_inset Formula $2\times2$ \end_inset real matrix \begin_inset Formula $A$ \end_inset can be written as \begin_inset Formula \begin{align} A & =a_{0}I+\bar{a}\cdot\bar{\sigma}\\ & =a_{0}I+a_{x}\sigma_{x}+a_{y}\sigma_{y}+a_{z}\sigma_{z} \end{align} \end_inset where \begin_inset Formula \begin{align} \bar{a} & =\left(a_{x},a_{y},a_{z}\right)\\ \bar{\sigma} & =\left(\sigma_{x},\sigma_{y},\sigma_{z}\right) \end{align} \end_inset A further useful property can be obtained from the Einstein notation \begin_inset Formula \begin{align} \left(\bar{a}\cdot\bar{\sigma}\right)^{2} & =a_{j}\sigma_{j}a_{k}\sigma_{k}\\ & =a_{j}a_{k}\sigma_{j}\sigma_{k}\\ & =a_{j}a_{k}\left(\delta_{jk}I+i\epsilon_{jkl}\sigma_{l}\right)\\ & =a_{j}a_{k}\delta_{jk}I+ia_{j}a_{k}\epsilon_{jkl}\sigma_{l}\\ & =a_{j}a_{j}I+ia_{j}a_{k}\epsilon_{jkl}\sigma_{l} \end{align} \end_inset The second term term collapses to \begin_inset Formula $0$ \end_inset as \begin_inset Formula \begin{align} a_{j}a_{k}\epsilon_{jkl}\sigma_{l} & =\epsilon_{jkl}a_{j}a_{k}\sigma_{l}\\ & =\epsilon_{kjl}a_{k}a_{j}\sigma_{l}\text{ by swapping order of summation}\\ & =-\epsilon_{jkl}a_{k}a_{j}\sigma_{l}\text{ by permutation of \ensuremath{\epsilon_{jkl}}}\\ & =-\epsilon_{jkl}a_{j}a_{k}\sigma_{l}\text{ by commutativity of scalar multiplication}\\ & \rightarrow\epsilon_{jkl}a_{j}a_{k}\sigma_{l}=-\epsilon_{jkl}a_{j}a_{k}\sigma_{l}=0 \end{align} \end_inset Thus, \begin_inset Formula \begin{align} \left(\bar{a}\cdot\bar{\sigma}\right)^{2} & =a_{j}a_{j}I\label{eq:magnitude pauli vector} \end{align} \end_inset \end_layout \begin_layout Subsection Calculating the matrix exponential \begin_inset Formula $e^{At}$ \end_inset \begin_inset CommandInset label LatexCommand label name "subsec:Calculating-the-matrix" \end_inset \end_layout \begin_layout Standard We can decompose \begin_inset Formula $A$ \end_inset from Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:matrixA" plural "false" caps "false" noprefix "false" \end_inset ) in terms of Pauli Matrices as \begin_inset Formula \begin{equation} A=-\frac{\Omega}{2Q}I+\frac{\Omega}{2Q}\left(Q^{2}-1\right)\sigma_{x}-\frac{i\Omega}{2Q}\left(Q^{2}+1\right)\sigma_{y}-\frac{\Omega}{2Q}\sigma_{z} \end{equation} \end_inset or \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} A=a_{0}I+\bar{a}\cdot\bar{\sigma} \end{equation} \end_inset where \begin_inset Formula \begin{equation} \begin{aligned}\bar{a} & =\left(\frac{\Omega}{2Q}\left(Q^{2}-1\right),-\frac{i\Omega}{2Q}\left(Q^{2}+1\right),-\frac{\Omega}{2Q}\right)\\ a_{0} & =-\frac{\Omega}{2Q} \end{aligned} \label{eq:A_vectors} \end{equation} \end_inset For our convenience later on, we rewrite \begin_inset Formula $\bar{a}$ \end_inset to get \begin_inset Formula \begin{equation} A=a_{0}I+\bar{a}\cdot\bar{\sigma}=a_{0}I+c_{a}\hat{a}\cdot\bar{\sigma}\label{eq:rewrite abar} \end{equation} \end_inset where we require \begin_inset Formula $\hat{a}$ \end_inset to be a unit vector such that \begin_inset Formula \begin{equation} \left(\hat{a}\cdot\bar{\sigma}\right)^{2}=I \end{equation} \end_inset Using Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:magnitude pauli vector" plural "false" caps "false" noprefix "false" \end_inset ) and Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:rewrite abar" plural "false" caps "false" noprefix "false" \end_inset ), we can calculate \begin_inset Formula $c_{a}$ \end_inset as \begin_inset Formula \begin{align} \left(\bar{a}\cdot\bar{\sigma}\right)^{2} & =c_{a}^{2}\left(\hat{a}\cdot\bar{\sigma}\right)^{2}\\ a_{j}a_{j}I & =c_{a}^{2}I\\ c_{a} & =\sqrt{a_{j}a_{j}}\\ & =\sqrt{\left[\frac{\Omega}{2Q}\left(Q^{2}-1\right)\right]^{2}+\left[-\frac{i\Omega}{2Q}\left(Q^{2}+1\right)\right]^{2}+\left(-\frac{\Omega}{2Q}\right)^{2}}\\ & =\frac{\Omega}{2Q}\sqrt{1-4Q^{2}}\label{eq:ca} \end{align} \end_inset Then \begin_inset Formula \begin{align} \hat{a} & =\frac{1}{c_{a}}\bar{a}\\ & =\frac{1}{\frac{\Omega}{2Q}\sqrt{1-4Q^{2}}}\left(\frac{\Omega}{2Q}\left(Q^{2}-1\right),-\frac{i\Omega}{2Q}\left(Q^{2}+1\right),-\frac{\Omega}{2Q}\right)\\ & =\frac{1}{\sqrt{1-4Q^{2}}}\left(Q^{2}-1,-i\left(Q^{2}+1\right),-1\right) \end{align} \end_inset so that \begin_inset Formula \begin{align} \hat{a}\cdot\bar{\sigma} & =\frac{1}{\sqrt{1-4Q^{2}}}\left(Q^{2}-1,-i\left(Q^{2}+1\right),-1\right)\cdot\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)\\ & =\frac{1}{\sqrt{1-4Q^{2}}}\left[\left(Q^{2}-1\right)\left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right)-i\left(Q^{2}+1\right)\left(\begin{array}{cc} 0 & -i\\ i & 0 \end{array}\right)-1\left(\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right)\right]\\ & =\frac{1}{\sqrt{1-4Q^{2}}}\left(\begin{array}{cc} -1 & -2\\ 2Q^{2} & 1 \end{array}\right)\label{eq:normalised pauli vector} \end{align} \end_inset Using Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:rewrite abar" plural "false" caps "false" noprefix "false" \end_inset ) and Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:normalised pauli vector" plural "false" caps "false" noprefix "false" \end_inset ), we can now calculate the matrix exponential \end_layout \begin_layout Standard \begin_inset Formula \begin{align} e^{At} & =\exp\left(ta_{0}I+t\bar{a}\cdot\bar{\sigma}\right)\\ & =\exp\left(ta_{0}\right)\exp\left(t\bar{a}\cdot\bar{\sigma}\right)\\ & =\exp\left(ta_{0}\right)\exp\left(tc_{a}\hat{a}\cdot\bar{\sigma}\right)\\ & =\exp\left(ta_{0}\right)\sum_{k=0}^{\infty}\frac{1}{k!}\left(tc_{a}\hat{a}\cdot\bar{\sigma}\right)^{k}\\ & =\exp\left(ta_{0}\right)\left\{ \sum_{n=0}^{\infty}\frac{1}{\left(2n\right)!}\left(tc_{a}\hat{a}\cdot\bar{\sigma}\right)^{2n}+\sum_{n=0}^{\infty}\frac{1}{\left(2n+1\right)!}\left(tc_{a}\hat{a}\cdot\bar{\sigma}\right)^{2n+1}\right\} \\ & =\exp\left(ta_{0}\right)\left\{ \sum_{n=0}^{\infty}\frac{1}{\left(2n\right)!}\left(tc_{a}\right)^{2n}\left(\hat{a}\cdot\bar{\sigma}\right)^{2n}+\sum_{n=0}^{\infty}\frac{1}{\left(2n+1\right)!}\left(tc_{a}\right)^{2n+1}\left(\hat{a}\cdot\bar{\sigma}\right)^{2n}\left(\hat{a}\cdot\bar{\sigma}\right)\right\} \\ & =\exp\left(ta_{0}\right)\left\{ \sum_{n=0}^{\infty}\frac{1}{\left(2n\right)!}\left(tc_{a}\right)^{2n}I^{n}+\sum_{n=0}^{\infty}\frac{1}{\left(2n+1\right)!}\left(tc_{a}\right)^{2n+1}I^{n}\left(\hat{a}\cdot\bar{\sigma}\right)\right\} \\ & =\exp\left(ta_{0}\right)\left\{ I\sum_{n=0}^{\infty}\frac{1}{\left(2n\right)!}\left(tc_{a}\right)^{2n}+\left(\hat{a}\cdot\bar{\sigma}\right)\sum_{n=0}^{\infty}\frac{1}{\left(2n+1\right)!}\left(tc_{a}\right)^{2n+1}\left(\hat{a}\cdot\bar{\sigma}\right)\right\} \\ & =\exp\left(ta_{0}\right)\left\{ I\cosh\left(tc_{a}\right)+\left(\hat{a}\cdot\bar{\sigma}\right)\sinh\left(tc_{a}\right)\right\} \end{align} \end_inset We can substitute from \begin_inset CommandInset ref LatexCommand ref reference "eq:normalised pauli vector" plural "false" caps "false" noprefix "false" \end_inset and \begin_inset CommandInset ref LatexCommand ref reference "eq:ca" plural "false" caps "false" noprefix "false" \end_inset to get the matrix exponential in the overdamped regime \begin_inset Formula \begin{equation} \begin{aligned}e^{At} & =e^{-\frac{\Omega}{2Q}t}\cosh\left(t\frac{\Omega}{2Q}\sqrt{1-4Q^{2}}\right)\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right)\\ & +e^{-\frac{\Omega}{2Q}t}\frac{1}{\sqrt{1-4Q^{2}}}\sinh\left(t\frac{\Omega}{2Q}\sqrt{1-4Q^{2}}\right)\left(\begin{array}{cc} -1 & -2\\ 2Q^{2} & 1 \end{array}\right) \end{aligned} \label{eq:exponential_overdamped} \end{equation} \end_inset In the underdamped regime, the \begin_inset Formula $\sinh$ \end_inset and \begin_inset Formula $\cosh$ \end_inset terms switch over to \begin_inset Formula $\sin$ \end_inset and \begin_inset Formula $\cos$ \end_inset terms, as \begin_inset Formula \begin{align} \sinh x & =-i\sin\left(ix\right)\\ \cosh x & =\cos\left(ix\right) \end{align} \end_inset yielding \begin_inset Formula \begin{equation} \begin{aligned}e^{At} & =e^{-\frac{\Omega}{2Q}t}\cos\left(t\frac{\Omega}{2Q}\sqrt{4Q^{2}-1}\right)\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right)\\ & +e^{-\frac{\Omega}{2Q}t}\frac{1}{\sqrt{4Q^{2}-1}}\sin\left(t\frac{\Omega}{2Q}\sqrt{4Q^{2}-1}\right)\left(\begin{array}{cc} -1 & -2\\ 2Q^{2} & 1 \end{array}\right) \end{aligned} \label{eq:exponential_underdamped} \end{equation} \end_inset \end_layout \begin_layout Paragraph A note on convergence \end_layout \begin_layout Standard At first glance, the second term in Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:exponential_overdamped" plural "false" caps "false" noprefix "false" \end_inset ) and Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:exponential_underdamped" plural "false" caps "false" noprefix "false" \end_inset ) respectively may have a convergence issue as \begin_inset Formula $Q\rightarrow\frac{1}{2}$ \end_inset and \begin_inset Formula $\pm\left(1-4Q^{2}\right)\rightarrow0$ \end_inset . We can doublecheck this by calculating the limit of \begin_inset Formula $\frac{\sinh\left(rx\right)}{x}$ \end_inset as \begin_inset Formula $x\rightarrow\infty$ \end_inset where \begin_inset Formula $r=t\frac{\Omega}{2Q}$ \end_inset . \begin_inset Formula \begin{align} \underset{x\rightarrow0}{\lim}\frac{\sinh\left(rx\right)}{x} & =\underset{x\rightarrow0}{\lim}\frac{1}{x}\sum_{n=0}^{\infty}\frac{1}{\left(2n+1\right)!}\left(rx\right)^{2n+1}\\ & =\underset{x\rightarrow0}{\lim}\sum_{n=0}^{\infty}\frac{r^{2n+1}x^{2n}}{\left(2n+1\right)!}\\ & =\underset{x\rightarrow0}{\lim}\left[r+\frac{r^{3}x^{2}}{6}+...\right]\\ & =r \end{align} \end_inset and likewise for \begin_inset Formula $\frac{\sin\left(rx\right)}{x}$ \end_inset . Thus, there is no issue with convergence, and the matrix exponential along the line of critical damping \begin_inset Formula $Q=\frac{1}{2}$ \end_inset is \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} e^{At}=e^{-\Omega t}\left\{ \left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right)+t\Omega\left(\begin{array}{cc} -1 & -2\\ \frac{1}{2} & 1 \end{array}\right)\right\} \end{equation} \end_inset \end_layout \begin_layout Section Stochastic models of Time-To-Live caching systems \end_layout \begin_layout Subsection Derivation of Poisson distribution from the Poisson process \begin_inset CommandInset label LatexCommand label name "subsec:Derivations-of-Poisson" \end_inset \end_layout \begin_layout Standard The Poisson process is defined by the transition probabilities \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{d}{dt}P_{0}\left(t\right) & =-\lambda_{U}P_{0}\left(t\right)\label{eq:poisson_process0}\\ \frac{d}{dt}P_{k+1}\left(t\right) & =\lambda_{U}P_{k}\left(t\right)-\lambda_{U}P_{k+1}\left(t\right)\,\,k>0\label{eq:poisson_processk}\\ P_{k}\left(0\right) & =\begin{cases} 1 & k=0\\ 0 & k>0 \end{cases} \end{align} \end_inset Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:poisson_process0" plural "false" caps "false" noprefix "false" \end_inset ) has solution \begin_inset Formula $P_{0}\left(t\right)=e^{-\lambda_{U}t}$ \end_inset ; the probability \begin_inset Formula $P\left[N_{U}\left(t\right)=0\right]$ \end_inset decays exponentially with constant rate \begin_inset Formula $\lambda_{U}$ \end_inset as time evolves. More generally, if we assume \begin_inset Formula \[ P_{k}\left(t\right)=f\left(t,k\right)e^{-\lambda_{U}t} \] \end_inset for some function \begin_inset Formula $f$ \end_inset and substitute into Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:poisson_processk" plural "false" caps "false" noprefix "false" \end_inset ), we get \begin_inset Formula \begin{align} \frac{d}{dt}\left[f\left(t,k+1\right)e^{-\lambda_{U}t}\right] & =\lambda_{U}\left[f\left(t,k\right)e^{-\lambda_{U}t}\right]-\lambda_{U}\left[f\left(t,k+1\right)e^{-\lambda_{U}t}\right]\\ e^{-\lambda_{U}t}\left[f'\left(t,k+1\right)-\lambda_{U}f\left(t,k+1\right)\right] & =e^{-\lambda_{U}t}\left[\lambda_{U}f\left(t,k\right)-\lambda_{U}f\left(t,k+1\right)\right]\\ f'\left(t,k+1\right) & =\lambda_{U}f\left(t,k\right) \end{align} \end_inset Hence, \begin_inset Formula $f\left(t,k,\lambda_{U}\right)=C\left(k\right)\left(\lambda_{U}t\right)^{k}$ \end_inset is polynomial in \begin_inset Formula $\lambda_{U}t$ \end_inset so that \begin_inset Formula \begin{align} \frac{d}{dt}\left[C\left(k+1\right)\left(\lambda_{U}t\right)^{k+1}\right] & =\lambda_{U}C\left(k\right)\left(\lambda_{U}t\right)^{k}\\ \left(k+1\right)\lambda_{U}C\left(k+1\right)\left(\lambda_{U}t\right)^{k} & =\lambda_{U}C\left(k\right)\left(\lambda_{U}t\right)^{k}\\ C\left(k+1\right) & =\frac{C\left(k\right)}{\left(k+1\right)}\\ & \Rightarrow C\left(k\right)=\frac{1}{k!} \end{align} \end_inset yielding \begin_inset Formula \begin{equation} P_{k}\left(t\right)=\frac{\left(\lambda_{U}t\right)^{k}e^{-\lambda_{U}t}}{k!}\label{eq:poisson_distribution} \end{equation} \end_inset Eq. ( \begin_inset CommandInset ref LatexCommand ref reference "eq:poisson_distribution" plural "false" caps "false" noprefix "false" \end_inset ) is the \emph on probability mass function \emph default for the Poisson distribution. For constant \begin_inset Formula $t=T$ \end_inset , if \begin_inset Formula \begin{equation} P\left[N_{U}\left(T\right)=k\right]=\frac{\left(\lambda_{U}T\right)^{k}e^{-\lambda_{U}T}}{k!} \end{equation} \end_inset then \begin_inset Formula $N_{U}\left(T\right)$ \end_inset follows a Poisson distribution \begin_inset CommandInset citation LatexCommand cite key "gardiner2009stochastic" literal "false" \end_inset with rate \begin_inset Formula $\lambda_{U}T$ \end_inset ; that is, \begin_inset Formula \begin{equation} N_{U}\left(T\right)\sim\text{Poi}\left(\lambda_{U}T\right) \end{equation} \end_inset \end_layout \begin_layout Subsection Expected value of Poisson distribution \begin_inset CommandInset label LatexCommand label name "subsec:Exp_value_poisson" \end_inset \end_layout \begin_layout Standard If \begin_inset Formula \begin{equation} X\sim\text{Poi}\left(\lambda\right) \end{equation} \end_inset then the expected value of \begin_inset Formula $X$ \end_inset is given by \begin_inset Formula \begin{align} \left\langle X\right\rangle & =\sum_{k=0}^{\infty}k\frac{\lambda^{k}e^{-\lambda}}{k!}\\ & =\lambda e^{-\lambda}\sum_{k=1}^{\infty}\frac{\lambda^{k-1}}{\left(k-1\right)!}\\ & =\lambda e^{-\lambda}e^{\lambda}\\ & =\lambda \end{align} \end_inset \end_layout \begin_layout Subsection Inter-event times of a Poisson process \begin_inset CommandInset label LatexCommand label name "subsec:Inter-event-times" \end_inset \end_layout \begin_layout Standard The inter-event times of a Poisson process with rate \begin_inset Formula $\lambda$ \end_inset are independent \begin_inset Formula $\text{Exp}\left(\lambda\right)$ \end_inset random variables. \end_layout \begin_layout Paragraph Proof \end_layout \begin_layout Standard Consider the case where a Poisson process is in the state \begin_inset Formula $N\left(T\right)=m$ \end_inset at some time \begin_inset Formula $T$ \end_inset . Let \begin_inset Formula $t_{0}>0$ \end_inset be the inter-event time, or the time until the next event occurs; that is, \begin_inset Formula $N\left(T+t_{0}\right)=m+1$ \end_inset and \begin_inset Formula $N\left(T+t\right)=m$ \end_inset for \begin_inset Formula $tx+y|X>y\right]=P\left[X>x\right] \end{equation} \end_inset for \begin_inset Formula $x,y\geq0$ \end_inset . This can be seen by considering \begin_inset Formula \begin{align} P\left[X>x+y|X>y\right] & =\frac{P\left[X>x+y,X>y\right]}{P\left[X>y\right]}\\ & =\frac{P\left[X>x+y\right]}{P\left[X>y\right]}\\ & =\frac{e^{-\lambda\left(x+y\right)}}{e^{-\lambda y}}\\ & =e^{-\lambda x}\\ & =P\left[X>x\right] \end{align} \end_inset \end_layout \begin_layout Subsection Expected value of shifted Exponential distribution \begin_inset CommandInset label LatexCommand label name "subsec:Exp_shifted_exponential" \end_inset \end_layout \begin_layout Standard If a random variable \begin_inset Formula $X$ \end_inset follows a shifted Exponential distribution \begin_inset Formula $X\sim\text{Exp}\left(\lambda,\tau\right)$ \end_inset , then \begin_inset Formula $X\in\left(\tau,\infty\right)$ \end_inset and \begin_inset Formula \begin{equation} \left\langle X\right\rangle =\intop_{s=\tau}^{s=\infty}s\lambda e^{-\lambda\left(s-\tau\right)}ds \end{equation} \end_inset Substitute \begin_inset Formula $r=s-\tau$ \end_inset . Then \begin_inset Formula \begin{align} \left\langle X\right\rangle & =\intop_{s=\tau}^{s=\infty}s\lambda e^{-\lambda\left(s-\tau\right)}ds\\ & \intop_{r=0}^{r=\infty}\left(r+\tau\right)\lambda e^{-\lambda r}dr\\ & =\intop_{0}^{\infty}r\lambda e^{-\lambda r}dr+\intop_{0}^{\infty}\tau\lambda e^{-\lambda r}dr \end{align} \end_inset Let \begin_inset Formula $u=r$ \end_inset and \begin_inset Formula $dv=\lambda e^{-\lambda r}dr$ \end_inset , then by parts \begin_inset Formula \begin{align} \left\langle X\right\rangle & =\left[-re^{-\lambda r}\right]_{0}^{\infty}-\intop_{0}^{\infty}\left(-e^{-\lambda r}\right)dr+\intop_{0}^{\infty}\tau\lambda e^{-\lambda r}dr\\ & =\left[-0+0\right]-\left[\frac{1}{\lambda}e^{-\lambda r}\right]_{0}^{\infty}+\left[-\tau e^{-\lambda r}\right]_{0}^{\infty}\\ & =-\left(0-\frac{1}{\lambda}\right)+\left[0-\left(-\tau\right)\right]\\ & =\frac{1}{\lambda}+\tau \end{align} \end_inset \end_layout \begin_layout Subsection Sum of Poisson random variables \begin_inset CommandInset label LatexCommand label name "subsec:Sum-of-Poisson" \end_inset \end_layout \begin_layout Standard If \begin_inset Formula $X\sim\text{Poi}\left(\lambda\right)$ \end_inset and \begin_inset Formula $Y\sim\text{Poi}\left(\mu\right)$ \end_inset are independent Poisson random variables, then their sum \begin_inset Formula $X+Y\sim\text{Poi}\left(\lambda+\mu\right)$ \end_inset \begin_inset CommandInset citation LatexCommand cite key "hogg1995introduction" literal "false" \end_inset . \end_layout \begin_layout Paragraph Proof \end_layout \begin_layout Standard If \begin_inset Formula $X\sim\text{Poi}\left(\lambda\right)$ \end_inset , then \begin_inset Formula $X$ \end_inset has probability mass function \begin_inset Formula \[ P\left[X=k\right]=\frac{\lambda^{k}e^{-\lambda}}{k!} \] \end_inset This can be proven by noting that if \begin_inset Formula $X\sim\text{Poi}\left(\lambda\right)$ \end_inset and \begin_inset Formula $Y\sim\text{Poi}\left(\mu\right)$ \end_inset are independent Poisson random variables, then their joint probability mass function is \begin_inset Formula \begin{align*} P\left[X=x,Y=y\right] & =P\left[X=x\right]P\left[Y=y\right]\\ & =\frac{\lambda^{x}e^{-\lambda}}{x!}\frac{\lambda^{y}e^{-\lambda}}{y!} \end{align*} \end_inset Then the probability mass function of the sum \begin_inset Formula $X+Y$ \end_inset is \begin_inset Formula \begin{align*} P\left[X+Y=k\right] & =\sum_{i=0}^{k}P\left[X=k-i,Y=i\right]\\ & =\sum_{i=0}^{k}\frac{\lambda^{k-i}e^{-\lambda}}{\left(k-i\right)!}\frac{\mu^{i}e^{-\mu}}{i!}\\ & =\frac{e^{-\lambda}e^{-\mu}}{k!}\sum_{i=0}^{k}\frac{k!}{\left(k-i\right)!i!}\lambda^{k-i}\mu^{i}\\ & =\frac{e^{-\left(\lambda+\mu\right)}}{k!}\sum_{i=0}^{k}\left(\begin{array}{c} k\\ i \end{array}\right)\lambda^{k-i}\mu^{i}\\ & =\frac{\left(\lambda+\mu\right)^{k}e^{-\left(\lambda+\mu\right)}}{k!} \end{align*} \end_inset yielding the probability mass function for a \begin_inset Formula $\text{Poi}\left(\lambda+\mu\right)$ \end_inset random variable. \end_layout \begin_layout Subsection Sum of Exponential random variables \begin_inset CommandInset label LatexCommand label name "subsec:Sum-of-Exponential" \end_inset \end_layout \begin_layout Standard If \begin_inset Formula $X_{i}\sim\text{Exp}\left(\lambda\right)$ \end_inset for \begin_inset Formula $i=1,2,...n$ \end_inset are independent random variables, then \begin_inset Formula \begin{equation} \sum_{i=1}^{n}X_{i}\sim\text{Gamma}\left(n,\lambda^{-1}\right) \end{equation} \end_inset \end_layout \begin_layout Paragraph Proof \end_layout \begin_layout Standard An Exponential \begin_inset Formula $\left(\lambda\right)$ \end_inset random variable \begin_inset Formula $X$ \end_inset has probability density function \begin_inset Formula \[ f_{X}\left(x,\lambda\right)=\lambda e^{-\lambda x} \] \end_inset such that \begin_inset Formula \[ P\left[X0$ \end_inset and scale parameter \begin_inset Formula $k>0$ \end_inset . If \begin_inset Formula $k=1$ \end_inset , then the Gamma distribution is identical to the Exponential distribution. If \begin_inset Formula $X\sim\text{Gamma}\left(\alpha,\lambda^{-1}\right)$ \end_inset and \begin_inset Formula $Y\sim\text{Gamma}\left(\beta,\lambda^{-1}\right)$ \end_inset are independent Gamma \begin_inset Formula $\left(k,\lambda^{-1}\right)$ \end_inset random variables, then their joint probability density function is \begin_inset Formula \[ f_{X,Y}\left(x,y,\alpha,\beta,\lambda\right)=f_{X}\left(x,\alpha,\lambda\right)f_{Y}\left(y,\beta,\lambda\right) \] \end_inset We can derive the probability density function for their sum \begin_inset Formula $Z=X+Y$ \end_inset by letting \begin_inset Formula $y=z-x$ \end_inset and integrating over \begin_inset Formula $x$ \end_inset : \begin_inset Formula \begin{align} f_{Z}\left(z\right) & =\intop_{0}^{z}f_{X}\left(x,\alpha,\lambda\right)f_{Y}\left(z-x,\beta,\lambda\right)dx\\ & =\intop_{0}^{z}\frac{\lambda^{\alpha}x^{\alpha-1}e^{-\lambda x}}{\Gamma\left(\alpha\right)}\frac{\lambda^{\beta}\left(z-x\right)^{\beta-1}e^{-\lambda\left(z-x\right)}}{\Gamma\left(\beta\right)}dx\\ & =\lambda^{\alpha}\lambda^{\beta}\intop_{0}^{z}\frac{x^{\alpha-1}e^{-\lambda x}}{\Gamma\left(\alpha\right)}\frac{\left(z-x\right)^{\beta-1}e^{-\lambda z}e^{\lambda x}}{\Gamma\left(\beta\right)}dx\\ & =\lambda^{\alpha+\beta}e^{-\lambda z}\intop_{x=0}^{x=z}\frac{x^{\alpha-1}\left(z-x\right)^{\beta-1}}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}dx \end{align} \end_inset Substitute \begin_inset Formula $x=zt$ \end_inset . Then \begin_inset Formula $dx=zdt$ \end_inset and \begin_inset Formula \begin{align} f_{Z}\left(z\right) & =\lambda^{\alpha+\beta}e^{-\lambda z}\intop_{t=0}^{t=1}\frac{\left(zt\right)^{\alpha-1}\left(z-zt\right)^{\beta-1}}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}zdt\\ & =\lambda^{\alpha+\beta}e^{-\lambda z}\intop_{0}^{1}\frac{z^{\alpha-1}t^{\beta-1}z^{\beta-1}\left(1-t\right)^{\beta-1}}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}zdt\\ & =\lambda^{\alpha+k_{2}}z^{\alpha+\beta-1}e^{-\lambda z}\intop_{0}^{1}\frac{t^{\alpha-1}\left(1-t\right)^{\beta-1}}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}dt\\ & =\frac{\lambda^{\alpha+\beta}z^{\alpha+\beta-1}e^{-\lambda z}}{\Gamma\left(\alpha+\beta\right)}\frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}\intop_{0}^{1}t^{\alpha-1}\left(1-t\right)^{\beta-1}dt\label{eq:integral} \end{align} \end_inset The integral in equation \begin_inset CommandInset ref LatexCommand ref reference "eq:integral" plural "false" caps "false" noprefix "false" \end_inset defines the Beta function \begin_inset CommandInset citation LatexCommand cite key "artin2015gamma" literal "false" \end_inset \begin_inset Formula \begin{equation} B\left(\alpha,\beta\right)=\intop_{0}^{1}t^{\alpha-1}\left(1-t\right)^{\beta-1}dt\label{eq:beta_function_integral} \end{equation} \end_inset By recursive integration by parts \begin_inset CommandInset citation LatexCommand cite key "aerin2020beta" literal "false" \end_inset , we can show that \begin_inset Formula \begin{equation} B\left(\alpha,\beta\right)=\frac{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}{\Gamma\left(\alpha+\beta\right)}\label{eq:beta_function_gamma} \end{equation} \end_inset Therefore, \begin_inset Formula \begin{equation} f_{Z}\left(z\right)=\frac{\lambda^{\alpha+\beta}z^{\alpha+\beta-1}e^{-\lambda z}}{\Gamma\left(\alpha+\beta\right)} \end{equation} \end_inset which is the probability density function for a Gamma \begin_inset Formula $\left(\alpha+\beta,\lambda^{-1}\right)$ \end_inset random variable. Hence, if \begin_inset Formula $X_{i}\sim\text{Gamma}\left(k_{i},\lambda^{-1}\right)$ \end_inset for \begin_inset Formula $i=1,2,...n$ \end_inset , then \begin_inset Formula \begin{equation} \sum_{i=1}^{n}X_{i}\sim\text{Gamma}\left(\sum_{i=1}^{n}k_{i},\lambda^{-1}\right) \end{equation} \end_inset If \begin_inset Formula $k_{i}=1$ \end_inset for each \begin_inset Formula $X_{i}$ \end_inset , then \begin_inset Formula $X_{i}\sim\text{Exp}\left(\lambda\right)$ \end_inset , and \begin_inset Formula \begin{equation} \sum_{i=1}^{n}X_{i}\sim\text{Gamma}\left(n,\lambda^{-1}\right) \end{equation} \end_inset \end_layout \begin_layout Subsection Variance of query process \begin_inset Formula $Q\left(t\right)$ \end_inset \begin_inset CommandInset label LatexCommand label name "subsec:Variance-of-query" \end_inset \end_layout \begin_layout Standard While the usage process \begin_inset Formula $U\left(t\right)$ \end_inset is well understood, being a straightforward Poisson process, we cannot say the same for \begin_inset Formula $Q\left(t\right)$ \end_inset . The inhibitory effect of caching which prevents multiple queries in rapid succession has a significant effect on the process. The process is a temporal equivalent to the Matern type-3 process, which is an inhibitory \emph on spacial \emph default point process in \begin_inset Formula $\mathbb{R}^{n}$ \end_inset , which was originally used to model the inhibitory effect that mature trees had on young trees growing nearby \begin_inset CommandInset citation LatexCommand cite key "huber2009likelihood,matern2013spatial" literal "false" \end_inset . These processes are known to quite intractable analytically, which makes calculating moments of \begin_inset Formula $N_{Q}\left(t\right)$ \end_inset quite difficult. In Subsection \begin_inset CommandInset ref LatexCommand ref reference "subsec:Query-and-record" plural "false" caps "false" noprefix "false" \end_inset , we derived the first moment (expected value) of \begin_inset Formula $N_{Q}\left(t\right)$ \end_inset \begin_inset Formula \begin{equation} \frac{\left\langle N_{Q}\left(t\right)\right\rangle }{t}=\frac{1}{\frac{1}{\lambda}+\tau} \end{equation} \end_inset by considering the ratio of the total time over the interquery time; this shortcut doesn't work for finding the second moment, which we need to obtain the variance. The probability mass function is somewhat intractable; calculating probabilitie s analytically is possible by recursive integration by parts, but calculating moments analytically is problematic, and will be left for further research. However, if we calculate the variance of the process by simulation, we observe that \begin_inset Formula \begin{equation} \frac{V\left[N_{Q}\left(t\right)\right]}{t}\simeq\frac{1}{\lambda^{2}\left(\frac{1}{\lambda}+\tau\right)^{3}} \end{equation} \end_inset as shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "fig:mean_variance" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \align center \begin_inset Graphics filename ../cache_dynamics/fig_query_process_mean_variance.pdf width 75text% \end_inset \begin_inset Caption Standard \begin_layout Plain Layout The mean and variance of \begin_inset Formula $N_{Q}\left(t\right)/t$ \end_inset appear to be approximately equal to analytic closed form expression for \begin_inset Formula $\lambda=10$ \end_inset , \begin_inset Formula $\tau=0.1$ \end_inset and \begin_inset Formula $t=100$ \end_inset . \begin_inset CommandInset label LatexCommand label name "fig:mean_variance" \end_inset \end_layout \end_inset \end_layout \end_inset This observation is by no means a proof of equality. However, the fact that both the mean and variance of \begin_inset Formula $N_{Q}\left(t\right)$ \end_inset appear to approach analytic closed form expression suggests that further analysis of this system would be worthwhile. \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash renewcommand{ \backslash leftmark}{Bibliography} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset bibtex LatexCommand bibtex btprint "btPrintCited" bibfiles "thesis" options "bibtotoc" \end_inset \end_layout \end_body \end_document