Power Law Behavior of Center-Like Decaying Oscillation : Exponent through Perturbation Theory and Optimization

TL;DR

This paper establishes a universal power law exponent of 1/3 for decaying center-like solutions in dynamical systems using perturbation theory and numerical optimization, aiding in distinguishing system behaviors.

Contribution

It introduces a general rule for the power law decay exponent of 1/3 in center-like solutions, supported by analytical and numerical methods.

Findings

Decaying center-like solutions follow a power law with exponent 1/3

The exponent is consistent across various rhythmic conditions

The method helps differentiate dynamical behaviors in complex systems

Abstract

In dynamical systems theory, there is a lack of a straightforward rule to distinguish exact center solutions from decaying center-like solutions, as both require the damping force function to be zero [1, 2]. By adopting a multi-scale perturbative method, we have demonstrated a general rule for the decaying center-like power law behavior, characterized by an exponent of 1/3 . The investigation began with a physical question about the higher-order nonlinearity in a damping force function, which exhibits birhythmic and trirhythmic behavior under a transition to a decaying center-type solution. Using numerical optimization algorithms, we identified the power law exponent for decaying center-type behavior across various rhythmic conditions. For all scenarios, we consistently observed a decaying power law with an exponent of 1/3 .Our study aims to elucidate their dynamical differences, contributing to theoretical insights and practical applications where distinguishing between different types of center-like behaviour is crucial. This key result would be beneficial for studying the multi-rhythmic nature of biological and engineering systems.