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Wavelet phase synchronization and chaoticity.

E B Postnikov¹

¹Department of Theoretical Physics, Kursk State University, 305000 Kursk, Russia. postnicov@gmail.com

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|April 7, 2010

Summary

Wavelet phase synchronization actually synchronizes smoothed chaotic signals, not the fine chaotic details. This method reveals averaged motion synchronization, offering new insights beyond classical chaotic phase synchronization.

Area of Science:

Nonlinear Dynamics and Chaos Theory
Signal Processing
Time-Series Analysis

Background:

Traditional chaotic phase synchronization focuses on fine-grained signal dynamics.
Wavelet phase synchronization, also known as time-scale synchronization, has been proposed as a method for analyzing chaotic signals.
The underlying mechanism of wavelet phase synchronization requires clarification.

Purpose of the Study:

To investigate the actual phenomenon captured by wavelet phase synchronization.
To elucidate the mathematical and topological underpinnings of this synchronization method.
To differentiate wavelet phase synchronization from classical chaotic phase synchronization.

Main Methods:

Representing the wavelet transform (Morlet wavelet) as a solution to a Cauchy problem for a diffusion equation.

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Analyzing the initial condition of the diffusion equation as a modulated harmonic function.

Topological analysis of the synchronization effect.

Main Results:

Wavelet phase synchronization is demonstrated to be the synchronization of smoothed functions, not the original chaotic signals.
The method effectively reduces chaotic fluctuations by focusing on underlying averaged dynamics.
The wavelet transform acts as a diffusion process, filtering out fine chaotic details.

Conclusions:

Wavelet phase synchronization reflects the synchronization of averaged dynamics, represented by bounding tori.
This differs from classical chaotic phase synchronization, which targets fine-level chaotic dynamics.
The study clarifies the nature of wavelet phase synchronization and its topological basis.