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Effect of smoothing on robust chaos
Amogh Deshpande1, Qingfei Chen, Yan Wang
1School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 28, 2010
Summary
Smoothing piecewise-smooth dynamical systems can introduce periodic windows, breaking robust chaos. The measure of these windows scales linearly with the smoothing parameter, offering new insights into chaos theory.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Mathematical Physics
Background:
- Robust chaos describes systems with only chaotic attractors in parameter intervals.
- This phenomenon is often linked to border-collision bifurcations in piecewise-smooth systems.
- Understanding transitions from robust chaos is crucial for predicting system behavior.
Purpose of the Study:
- To investigate the impact of smoothing on robust chaos in piecewise-smooth dynamical systems.
- To determine if smoothing can induce periodic behavior (windows) within regions of robust chaos.
- To quantify the relationship between the degree of smoothing and the emergence of periodic windows.
Main Methods:
- Introduction of a smoothing parameter into piecewise-smooth dynamical equations.
- Analysis of system attractors and bifurcations under varying degrees of smoothing.
- Development of a heuristic theory to explain observed scaling relations.
- Numerical simulations and experimental validation using electronic circuits.
Main Results:
- Smoothing piecewise-smooth systems can lead to the appearance of periodic windows.
- The measure of these periodic windows in parameter space scales linearly with the smoothing parameter.
- This linear scaling is independent of the specific smoothing function used.
- Experimental results confirm the emergence of periodic windows in a smoothed piecewise linear system.
Conclusions:
- Smoothing is an effective method for disrupting robust chaos and introducing periodic behavior.
- The linear scaling law provides a predictable relationship between smoothing and the prevalence of periodic windows.
- Findings have implications for understanding and controlling complex dynamics in physical and engineering systems.
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