Conley index for random dynamical systems

Abstract

Conley index theory is a very powerful tool in the study of dynamical systems. In this paper, we generalize Conley index theory to discrete random dynamical systems. Our constructions are basically the random version of Franks and Richeson in [J. Franks, D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352 (2000) 3305-3322] for maps, and the relations of isolated invariant sets between time-continuous random dynamical systems and corresponding time-h maps are discussed. Two examples are presented to illustrate results in this paper.

The random case of Conley's theorem: III. Random semiflow case and Morse decomposition

“…Both of these two theorems are originated from Conley [9]. My other two related (joint) works are [27,28].…”

A sufficient condition for bifurcation in random dynamical systems

“…Denote the corresponding discrete-time RDS by ϕ λ (n, ω, x). From Theorem 7.2 of [12] we know that {0} is the random isolated invariant set of ϕ λ (n, ω, x). If lim…”

“…We first recall some basic definitions from [12]. A random compact set N (ω) is called a random isolating neighborhood if it satisfies…”

α-limit sets and Lyapunov function for maps with one topological attractor

“…Conley [4] obtained the fundamental decomposition theorem of isolated invariant sets and extended it to Morse decomposition. Since then the Conley index theory had been studied extensively [2,18,16,24]. Liu [18] studied the Conley index for random dynamical systems, Wang [24] studied the Conley index and shape Conley index in general metric spaces.…”

“…Since then the Conley index theory had been studied extensively [2,18,16,24]. Liu [18] studied the Conley index for random dynamical systems, Wang [24] studied the Conley index and shape Conley index in general metric spaces. Here we consider the map f ∈ F on the compact metric space X.…”