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

Sandip Saha

TL;DR

The paper investigates how decaying center-like solutions arise in weakly nonlinear oscillators with higher-order nonlinear damping and seeks a universal decay rule distinguishing center-like from true centers. Using the Krylov–Bogoliubov averaging within a multi-scale perturbation framework, it derives slow amplitude equations for Van der Pol and Rayleigh-type systems, revealing a $t^{-1/3}$ decay law for center-like states after perturbations. Numerical optimization confirms the exponent across monorhythmic, birhythmic, and trirhythmic damping regimes, even when analytic solutions are intractable for generalized damping. This finding provides a practical, broadly applicable rule for center-like decay in multi-rhythmic biological and engineering contexts and points to future work on including higher-order nonlinearities and forcing effects.

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.

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

TL;DR

The paper investigates how decaying center-like solutions arise in weakly nonlinear oscillators with higher-order nonlinear damping and seeks a universal decay rule distinguishing center-like from true centers. Using the Krylov–Bogoliubov averaging within a multi-scale perturbation framework, it derives slow amplitude equations for Van der Pol and Rayleigh-type systems, revealing a decay law for center-like states after perturbations. Numerical optimization confirms the exponent across monorhythmic, birhythmic, and trirhythmic damping regimes, even when analytic solutions are intractable for generalized damping. This finding provides a practical, broadly applicable rule for center-like decay in multi-rhythmic biological and engineering contexts and points to future work on including higher-order nonlinearities and forcing effects.

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.

Paper Structure

This paper contains 5 sections, 10 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Description of the Problem
  3. Multi-scale Perturbative Method: a Tool to Reformulate a Question into a Mathematical Framework
  4. Algorithm and Corresponding Numerical Results
  5. Conclusions and Future Directions

Figures (4)

  • Figure 1: Numerical solution and fitted functions for different initial conditions for the Birhythmic case (VdP class) $(\alpha=0.144,~\beta=0.005)$: $r_0=$3.16 (top) and $r_0=$3.81 (bottom).
  • Figure 2: Numerical solution and fitted functions for different initial conditions for the Monorhythmic case (VdP class) $(\alpha=\beta=\gamma=\delta=0)$: $r_0=$3.97 (top) and $r_0=$4.23 (bottom).
  • Figure 3: Numerical solution and fitted functions for different initial conditions for the Birhythmic case (Rayleigh class) $(\alpha=0.285272,~\beta=0.0244993)$: $r_0=$1.77 (top) and $r_0=$2.51 (bottom).
  • Figure 4: Numerical solution and fitted functions for different initial conditions for the Monorhythmic case (Rayleigh class) $(\alpha=\beta=\gamma=\delta=0)$: $r_0=$2.0 (top) and $r_0=$2.23 (bottom).