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Transitions from phase-locked dynamics to chaos in a piecewise-linear map
Zhanybai T Zhusubaliyev1, Erik Mosekilde, Soma De
1Kursk State Technical University, Department of Computer Science, 50 Years of October Street, 94, Kursk 305040, Russia. zhanybai@mail.kursk.ru
Summary
This study explores torus destruction in piecewise-smooth maps, revealing novel transition-to-chaos mechanisms. These findings differ from prior models, offering new insights into complex dynamical systems.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Bifurcation Theory
Background:
- Torus formation in piecewise-smooth maps is linked to border-collision bifurcations.
- Previous research identified specific bifurcations for ergodic-to-resonant torus transitions.
Purpose of the Study:
- To investigate the transition to chaos via torus destruction in piecewise-smooth maps.
- To analyze the interplay of various bifurcations during this transition.
Main Methods:
- Analysis of a piecewise-linear normal-form map.
- Examination of border-collision, period-doubling, and homoclinic bifurcations.
Main Results:
- Demonstration of novel mechanisms for torus destruction leading to chaos.
- These mechanisms involve a complex interplay of different bifurcation types.
- Observed transition pathways qualitatively differ from established Afraimovich-Shilnikov scenarios.
Conclusions:
- The transition to chaos in these maps is more complex than previously understood.
- Border-collision bifurcations play a crucial role in novel chaos generation pathways.
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