Nonlinear oscillator with parametric colored noise: some analytical results

TL;DR

This paper analyzes the long-term behavior of a nonlinear oscillator driven by colored noise, developing a novel averaging method to account for noise-angle correlations and derive the system's probability distribution.

Contribution

It introduces a new averaging scheme for nonlinear oscillators with colored noise, overcoming limitations of previous white-noise methods.

Findings

Derived the probability distribution function of the system.

Calculated long-time behavior of physical observables.

Developed a specific averaging scheme for colored noise.

Abstract

The asymptotic behavior of a nonlinear oscillator subject to a multiplicative Ornstein-Uhlenbeck noise is investigated. When the dynamics is expressed in terms of energy-angle coordinates, it is observed that the angle is a fast variable as compared to the energy. Thus, an effective stochastic dynamics for the energy can be derived if the angular variable is averaged out. However, the standard elimination procedure, performed earlier for a Gaussian white noise, fails when the noise is colored because of correlations between the noise and the fast angular variable. We develop here a specific averaging scheme that retains these correlations. This allows us to calculate the probability distribution function (P.D.F.) of the system and to derive the behavior of physical observables in the long time limit.