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

2006/06/09 by Zhenxin Liu, Liu, Zhenxin, Shuguan Ji +3
Biochemistry, Genetics and Molecular Biology · Engineering · #37B25 #37B55 #37H99 #Dynamical Systems (math.DS) #FOS: Mathematics #RNA Research and Splicing #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.math/0606205

openalex publication_date 2006/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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