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A geometric theory of chaotic phase synchronization
Margaret Beck1, Kresimir Josić
1Department of Mathematics and Statistics and Center for BioDynamics, Boston University, Boston, Massachusetts 02215, USA.
Chaos (Woodbury, N.Y.)
|April 5, 2003
Summary
This study develops a mathematical framework for chaotic phase synchronization. It shows that chaotic systems with phase coherence may not always be treated as noisy periodic oscillators.
Area of Science:
- Nonlinear Dynamics
- Chaos Theory
- Mathematical Physics
Background:
- Chaotic phase synchronization is observed but lacks rigorous mathematical treatment.
- Existing theories often model chaotic systems as noisy periodic oscillators.
Purpose of the Study:
- Extend phase synchronization theory to chaotic systems.
- Develop a mathematical framework for phase coherent chaotic attractors.
- Investigate the role of attractor structure in chaotic phase synchronization.
Main Methods:
- Generalization of the Averaging Theorem for periodic systems.
- Extension of Kuramoto's geometric theory to chaotic oscillators.
- Analysis of special flows over diffeomorphisms with periodic perturbations.
Main Results:
- Reduced equations describing phase difference dynamics were derived.
- Demonstrated the importance of chaotic attractor structure in response to perturbations.
- Showed that chaotic phase coherent systems differ from noisy periodic oscillators.
Conclusions:
- A rigorous mathematical treatment for chaotic phase synchronization is presented.
- The study highlights the limitations of treating chaotic systems as simple noisy oscillators.
- The developed framework offers broader applicability to chaotic phase dynamics.
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