Typical dynamics and fluctuation analysis of slow–fast systems driven by fractional Brownian motion

Abstract

This paper studies typical dynamics and fluctuations for a slow–fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit rates of convergence, which may be of independent interest, we characterize the asymptotic dynamics of the slow component to two orders (i.e. the typical dynamics and the fluctuations). The limiting distribution of the fluctuations turns out to depend upon the manner in which the small-noise parameter is taken to zero relative to the scale-separation parameter. We study also an extension of the original model in which the relationship between the two small parameters leads to a qualitative difference in limiting behavior. The results of this paper provide an approximation, to two orders, to dynamical systems perturbed by small fractional Brownian noise and incorporating multiscale effects.