Random Attractor for Stochastic Hindmarsh-Rose Equations with Additive Noise

TL;DR

This paper investigates the long-term behavior of stochastic Hindmarsh-Rose equations with additive noise, establishing the existence of a random attractor and analyzing the system's pullback dynamics in a bounded domain.

Contribution

It proves the existence of a random attractor for stochastic Hindmarsh-Rose equations using uniform estimates and pullback dynamics analysis, advancing understanding of neurodynamic models under noise.

Findings

Existence of a random attractor for the stochastic system

Pullback absorbing and asymptotic compactness properties established

Long-term dynamics characterized in the $L^2$ Hilbert space

Abstract

For stochastic Hindmarsh-Rose equations with additive noises in the study of neurodynamics, the longtime and global pullback dynamics on a two-dimensional bounded domain is explored in this work. Using the additive transformation and by the sharp uniform estimates, we proved the pullback absorbing and the pullback asymptotically compact characteristics of the Hindmarsh-Rose random dynamical system in the $L^2$ Hilbert space. It shows the existence of a random attractor for this random dynamical system.