Stochastic LimitCycle Oscillations of a Nonlinear System Under Random Perturbations
Abstract
Dynamical systems with ε 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 timeinhomogeneous Gaussian process, near a deterministic limit cycle in R^{n}. Based on respectively the theory of random perturbations of dynamical systems and the WKB approximation that codes the large deviations principle (LDP), 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 LDP of diffusion processes. We study uniform asymptotic behavior of stochastic limit cycles through the interchange of limits of time t →∞ and ε →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 ε in the theory is justified by a novel scaling hypothesis via constructing a sequence of stochastic differential equations.
 Publication:

Journal of Statistical Physics
 Pub Date:
 March 2021
 DOI:
 10.1007/s10955021027242
 arXiv:
 arXiv:2009.03000
 Bibcode:
 2021JSP...182...47C
 Keywords:

 Stochastic limit cycles;
 Central limit theorem;
 Large deviation principle;
 Random perturbations of dynamical systems;
 WKB approximation;
 Entropy balance;
 Scaling hypothesis;
 Mathematical Physics;
 Mathematics  Dynamical Systems
 EPrint:
 doi:10.1007/s10955021027242