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Dynamics of bouncing convex body

Author

Listed:
  • Zhang, Xiaoming
  • Li, Denghui
  • Grebogi, Celso
  • Liu, Xianbin

Abstract

We consider a dynamical system that describes a convex planer body with a smooth boundary, which falls under the influence of gravity and is elastically reflected upon hitting a wall. Under suitable restriction on the system’s parameters, the system is equivalent to a billiard problem within a cylinder. We prove that the dynamics of the system is intimately linked to a variational problem, corresponding to a twist map. Moreover, we utilise the established variational method to investigate the stability of trivial periodic orbits and near-integrable dynamics when this planar body closely approximates a disc with its centre of mass at the geometric centre. Numerical results illustrates our mathematical findings and highlight the system’s intricate dynamics.

Suggested Citation

  • Zhang, Xiaoming & Li, Denghui & Grebogi, Celso & Liu, Xianbin, 2024. "Dynamics of bouncing convex body," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).
    • Handle: RePEc:eee:chsofr:v:183:y:2024:i:c:s0960077924004478
    DOI: 10.1016/j.chaos.2024.114895
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    References listed on IDEAS

    as
    1. A. Douglas Stone, 2010. "Chaotic billiard lasers," Nature, Nature, vol. 465(7299), pages 696-697, June.
    2. Li, Denghui & Zhang, Xiaoming & Liu, Xianbin & Xie, Jianhua & Grebogi, Celso, 2023. "Boundedness of solutions for a bouncing ball model with quasiperiodic moving wall," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    3. Zhang, Xiaoming & Xie, Jianhua & Li, Denghui & Cao, Zhenbang & Grebogi, Celso, 2022. "Stability analysis of the breathing circle billiard," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    Full references (including those not matched with items on IDEAS)

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