Universal generation of devil's staircases near Hopf bifurcations via modulated forcing of nonlinear systems

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Lingnau, Benjamin
Shortiss, Kevin
Dubois, Fabien
Peters, Frank H.
Kelleher, Bryan
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American Physical Society
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The discrete circle map is the archetypical example of a driven periodic system, showing a complex resonance structure under a change of the forcing frequency known as the devil's staircase. Adler's equation can be seen as the direct continuous equivalent of the circle map, describing locking effects in periodic systems with continuous forcing. This type of locking produces a single fundamental resonance tongue without higher-order resonances, and a devil's staircase is not observed. We show that, with harmonically modulated forcing, nonlinear oscillations close to a Hopf bifurcation generically reproduce the devil's staircase even in the continuous case. Experimental results on a semiconductor laser driven by a modulated optical signal show excellent agreement with our theoretical predictions. The locking appears as a modulation of the oscillation amplitude as well as the angular oscillation frequency. Our results show that by proper implementation of an external drive, additional regions of stable frequency locking can be introduced in systems which originally show only a single Adler-type resonance tongue. The induced resonances can be precisely controlled via the modulation parameters.
Locks (fasteners) , Modulation , Resonance , Stairs , Angular oscillations , Complex resonances , Forcing frequencies , Fundamental resonance , Higher order resonances , Modulation parameters , Nonlinear oscillation , Oscillation amplitude , Hopf bifurcation
Lingnau, B., Shortiss, K., Dubois, F., Peters, F. H. and Kelleher, B. (2020) 'Universal generation of devil's staircases near Hopf bifurcations via modulated forcing of nonlinear systems', Physical Review E, 102(3), 030201 (6pp). doi: 10.1103/PhysRevE.102.030201