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Neimark-Sacker bifurcations and evidence of chaos in a discrete dynamical model of walkers

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  • Rahman, Aminur
  • Blackmore, Denis

Abstract

Bouncing droplets on a vibrating fluid bath can exhibit wave-particle behavior, such as being propelled by interacting with its own wave field. These droplets seem to walk across the bath, and thus are dubbed walkers. Experiments have shown that walkers can exhibit exotic dynamical behavior indicative of chaos. While the integro-differential models developed for these systems agree well with the experiments, they are difficult to analyze mathematically. In recent years, simpler discrete dynamical models have been derived and studied numerically. The numerical simulations of these models show evidence of exotic dynamics such as period doubling bifurcations, Neimark–Sacker (N–S) bifurcations, and even chaos. For example, in [1], based on simulations Gilet conjectured the existence of a supercritical N-S bifurcation as the damping factor in his one- dimensional path model. We prove Gilet’s conjecture and more; in fact, both supercritical and subcritical (N-S) bifurcations are produced by separately varying the damping factor and wave-particle coupling for all eigenmode shapes. Then we compare our theoretical results with some previous and new numerical simulations, and find complete qualitative agreement. Furthermore, evidence of chaos is shown by numerically studying a global bifurcation.

Suggested Citation

  • Rahman, Aminur & Blackmore, Denis, 2016. "Neimark-Sacker bifurcations and evidence of chaos in a discrete dynamical model of walkers," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 339-349.
    • Handle: RePEc:eee:chsofr:v:91:y:2016:i:c:p:339-349
    DOI: 10.1016/j.chaos.2016.06.016
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    References listed on IDEAS

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    1. Blackmore, Denis & Rahman, Aminur & Shah, Jigar, 2009. "Discrete dynamical modeling and analysis of the R–S flip-flop circuit," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 951-963.
    2. Stéphane Perrard & Matthieu Labousse & Marc Miskin & Emmanuel Fort & Yves Couder, 2014. "Self-organization into quantized eigenstates of a classical wave-driven particle," Nature Communications, Nature, vol. 5(1), pages 1-8, May.
    3. Y. Couder & S. Protière & E. Fort & A. Boudaoud, 2005. "Walking and orbiting droplets," Nature, Nature, vol. 437(7056), pages 208-208, September.
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    Cited by:

    1. Li, Wen-na & Elsadany, A.A. & Zhou, Wei & Zhu, Yan-lan, 2021. "Global Analysis, Multi-stability and Synchronization in a Competition Model of Public Enterprises with Consumer Surplus," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).

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