Power law behavior of center-like decaying oscillation: Exponent through Perturbation Theory and Optimization

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 (Sarkar et al., 2011; Saha and Gangopadhyay, 2018). 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 $\frac{1}{3}$. 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 $\frac{1}{3}$. Our study aims to elucidate their dynamical differences, contributing to theoretical insights and practical applications where distinguishing between different types of center-like behavior is crucial. This key result would be beneficial for studying the multi-rhythmic nature of biological and engineering systems.