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Local bifurcation structure of a bouncing ball system with a piecewise polynomial function for table displacement
Yudai Okishio1, Hiroaki Ito1, Hiroyuki Kitahata1
1Department of Physics, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba 263-8522, Japan.
Chaos (Woodbury, N.Y.)
|September 3, 2020
Summary
The study reveals how table vibration patterns affect the chaotic dynamics of a bouncing ball system. Different polynomial vibration functions alter the system's bifurcation diagrams, impacting its predictable behavior.
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.
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