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Dynamical formation of stable irregular transients in discontinuous map systems
Hailin Zou1, Shuguang Guan, C-H Lai
1Department of Physics, National University of Singapore, Singapore, Singapore.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2009
Summary
Stable chaos, characterized by long irregular transients in high-dimensional dynamical systems, arises from trajectories repeatedly approaching and jumping from basin boundaries. This study reveals the hidden mechanism behind these dynamics in coupled discontinuous maps.
Area of Science:
- Dynamical Systems
- Chaos Theory
- Nonlinear Dynamics
Background:
- Stable chaos involves long, irregular transients in high-dimensional dynamical systems, often with a negative largest Lyapunov exponent.
- The underlying mechanisms of stable chaos, particularly the formation of irregular transients, remain poorly understood.
- Coupled discontinuous map systems offer a platform to explore complex dynamical behaviors.
Purpose of the Study:
- To investigate the dynamical formation of stable irregular transients in coupled discontinuous map systems.
- To elucidate the hidden patterns and mechanisms responsible for stable chaos.
- To provide a clear visualization and verification of the proposed formation process.
Main Methods:
- Analysis of coupled discontinuous map systems.
- Investigation of transient dynamics in phase space.
- Measurement of distance sequences between trajectories and basin boundaries.
- Numerical experiments to verify the proposed mechanism.
Main Results:
- Transient dynamics exhibit a hidden pattern of repeatedly approaching and jumping from a basin boundary.
- This pattern is visualized by analyzing distance sequences to the basin boundary.
- The formation of stable chaos is linked to the intersection points of discontinuous boundaries and their images.
Conclusions:
- The study elucidates the mechanism of stable chaos formation in coupled discontinuous map systems.
- A novel understanding of transient dynamics involving basin boundary interactions is presented.
- Numerical evidence supports the proposed origin of stable chaos from boundary intersection dynamics.
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