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Local bifurcation structure of a bouncing ball system with a piecewise polynomial function for table displacement.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Classical Mechanics

Background:

  • The bouncing ball system, a small rigid ball on a vibrating table, is a classic model in nonlinear dynamics.
  • Previous studies explored its behavior under various conditions, but the impact of specific vibration function types remained less understood.

Purpose of the Study:

  • To investigate how the order of piecewise polynomial functions used for table vibration influences the bifurcation diagrams of the bouncing ball system.
  • To elucidate the underlying mechanisms causing qualitative differences in these diagrams.

Main Methods:

  • Analysis of the two-period solution in the bouncing ball system.
  • Derivation of approximate curves near period-doubling bifurcation points for piecewise cubic table vibrations.
  • Numerical calculations to validate theoretical approximations.

Main Results:

  • The order of the polynomial function defining the table's vibration qualitatively alters the system's bifurcation diagram.
  • A detailed mechanism for these diagram differences was identified by focusing on the two-period solution.
  • Derived approximations accurately reproduced numerical results for piecewise cubic vibrations.

Conclusions:

  • The mathematical form of the table's vibration function is critical in determining the complex dynamics and bifurcations of the bouncing ball system.
  • The study provides a method for approximating system behavior near critical bifurcation points, enhancing predictability in chaotic systems.