Anomalous diffusion in nonlinear oscillators with multiplicative noise

TL;DR

This paper investigates the long-term behavior of nonlinear oscillators with random frequency, revealing algebraic growth in energy and position, and analyzing how noise correlations affect anomalous diffusion.

Contribution

It provides analytical and numerical insights into the scaling laws and diffusion constants for nonlinear oscillators under multiplicative noise, including effects of noise correlation.

Findings

Energy and position grow algebraically over time.

Correlated noise results in anomalous diffusion exponents half those of white noise.

Explicit calculations of scaling exponents and diffusion constants.

Abstract

The time-asymptotic behavior of undamped, nonlinear oscillators with a random frequency is investigated analytically and numerically. We find that averaged quantities of physical interest, such as the oscillator's mechanical energy, root-mean-square position and velocity, grow algebraically with time. The scaling exponents and associated generalized diffusion constants are calculated when the oscillator's potential energy grows as a power of its position. Correlated noise yields anomalous diffusion exponents equal to half the value found for white noise.