Stochastic limit-cycle oscillations of a nonlinear system under random perturbations

TL;DR

This paper analyzes how small random perturbations affect nonlinear systems near a deterministic limit cycle, connecting Gaussian fluctuations, large deviations, and physical interpretations through rigorous mathematical analysis.

Contribution

It provides a comprehensive analysis of stochastic limit cycles, linking CLT, large deviations, and physical properties, with a novel scaling hypothesis for the perturbation parameter.

Findings

Rigorous connection between Gaussian fluctuations and large deviation rate function curvature.

Asymptotic behavior of stochastic limit cycles as time approaches infinity and perturbation vanishes.

Characterizations include probability flux approximation, vector field features, and entropy balance along the cycle.

Abstract

Dynamical systems with $\epsilon$ small random perturbations appear in both continuous mechanical motions and discrete stochastic chemical kinetics. The present work provides a detailed analysis of the central limit theorem (CLT), with a time-inhomogeneous Gaussian process, near a deterministic limit cycle in $\mathbb{R}^n$. Based on the theory of random perturbations of dynamical systems and the WKB approximation respectively, results are developed in parallel from both standpoints of stochastic trajectories and transition probability density and their relations are elucidated. We show rigorously the correspondence between the local Gaussian fluctuations and the curvature of the large deviation rate function near its infimum, connecting the CLT and the large deviation principle of diffusion processes. We study uniform asymptotic behavior of stochastic limit cycles through the interchange of limits of time $t\to\infty$ and $\epsilon\to 0$. Three further characterizations of stochastic limit cycle oscillators are obtained: (i) An approximation of the probability flux near the cycle; (ii) Two special features of the vector field for the cyclic motion; (iii) A local entropy balance equation along the cycle with clear physical meanings. Lastly and different from the standard treatment, the origin of the $\epsilon$ in the theory is justified by a novel scaling hypothesis via constructing a sequence of stochastic differential equations.