Compactification for asymptotically autonomous dynamical systems: Theory, applications and invariant manifolds

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Wieczorek, Sebastian
Xie, Chun
Jones, Chris K.R.T.
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We develop a general compactification framework to facilitate analysis of nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant sets of the limit systems from infinity to analyze the original nonautonomous problem in the spirit of dynamical systems theory. Specifically, we prove that solutions of interest are contained in unique invariant manifolds of saddles for the limit systems when embedded in the extended phase space. The uniqueness holds in the general case, that is even if the compactification gives rise to a centre direction and the manifolds become centre or centre-stable manifolds. A wide range of problems including pullback attractors, rate-induced critical transitions (R-tipping) and nonlinear wave solutions fit naturally into our framework, and their analysis can be greatly simplified by the compactification.
Asymptotically autonomous dynamical systems , Compactification , Nonautonomous instabilities , Nonlinear waves , Rate-induced tipping
Wieczorek, S., Xie, C. and Jones, C. K.R.T. (2021) 'Compactification for asymptotically autonomous dynamical systems: Theory, applications and invariant manifolds', Nonlinearity, 34(5), pp. 2970-3000. doi: 10.1088/1361-6544/abe456
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© 2021, IOP Publishing Ltd & London Mathematical Society. This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at