Effects of parametric noise on a nonlinear oscillator

TL;DR

This paper investigates how parametric noise affects a nonlinear oscillator, deriving the probability distribution's asymptotic behavior and analyzing how different noise types influence the system's anomalous diffusion properties.

Contribution

It provides a detailed analysis of the effects of white and colored parametric noise on nonlinear oscillators, including the derivation of anomalous diffusion exponents and their dependence on noise correlation.

Findings

Algebraic growth of observables in small damping limit

Modification of exponents with colored noise

Quantitative characterization of noise influence on diffusion

Abstract

We study a model of a nonlinear oscillator with a random frequency and derive the asymptotic behavior of the probability distribution function when the noise is white. In the small damping limit, we show that the physical observables grow algebraically with time before the dissipative time scale is reached, and calculate the associated anomalous diffusion exponents. In the case of colored noise, with a nonzero but arbitrarily small correlation time, the characteristic exponents are modified. We determine their values thanks to a self-consistent Ansatz.