Attractor-repeller pair, Morse decomposition and Lyapunov function for random dynamical systems

TL;DR

This paper extends the concepts of attractor-repeller pairs, Morse decompositions, and Lyapunov functions to random dynamical systems, providing new characterizations that enhance understanding of stability in stochastic settings.

Contribution

It introduces stronger definitions of random attractors and repellers and characterizes their decompositions via Lyapunov functions in the context of random dynamical systems.

Findings

Characterization of attractor-repeller pairs through Lyapunov functions

Extension of Morse decomposition to random systems

New definitions of random attractors and repellers

Abstract

In the stability theory of dynamical systems, Lyapunov functions play a fundamental role. In this paper, we study the attractor-repeller pair decomposition and Morse decomposition for compact metric space in the random setting. In contrast to [8], by introducing slightly stronger definitions of random attractor and repeller, we characterize attractor-repeller pair decompositions and Morse decompositions for random dynamical systems through the existence of Lyapunov functions. These characterizations, we think, deserve to be known widely.