Scaling behavior of a nonlinear oscillator with additive noise, white and colored

TL;DR

This paper analyzes how a nonlinear oscillator's amplitude, velocity, and energy grow over time under additive noise, providing analytical and numerical insights for both white and colored noise scenarios.

Contribution

It offers the first analytical expressions for the energy distribution and anomalous diffusion exponents in a nonlinear oscillator with additive noise, including colored noise effects.

Findings

Energy grows algebraically with time.

Analytical energy distribution for white noise derived.

Anomalous diffusion exponents for colored noise calculated.

Abstract

We study analytically and numerically the problem of a nonlinear mechanical oscillator with additive noise in the absence of damping. We show that the amplitude, the velocity and the energy of the oscillator grow algebraically with time. For Gaussian white noise, an analytical expression for the probability distribution function of the energy is obtained in the long-time limit. In the case of colored, Ornstein-Uhlenbeck noise, a self-consistent calculation leads to (different) anomalous diffusion exponents. Dimensional analysis yields the qualitative behavior of the prefactors (generalized diffusion constants) as a function of the correlation time.