Decompositions and Liapounov functions

Huang, Tusen

doi:10.1016/s0960-0779(00)00253-8

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Introduction and Main Results1

Lemma 43 Assume A(ω) Is a Random Attractor And B(a)(ω) Is Th...1

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Cited by 6 publications

(2 citation statements)

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“…Among interesting results in [6] is a proposition which claims that mutually disjoint invariant sets are an attractor-repeller pair if and only if there exists a Lyapunov function for those sets. The result is then developed in Huang [9] as a criterion to detect Morse sets and Morse decomposition once a Lyapunov function for those sets can be constructed.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Attractor–repeller Pair, Morse Decomposition and Lyapunov Function for Random Dynamical Systems

Liu

2008

Stoch. Dyn.

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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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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof. The proof is completely similar to that of Lemma 6 in [9], so we omit the details here. ✷ Lemma 5.2 Assume D = {M 1 , M 2 , .…”

Section: Lemma 43 Assume A(ω) Is a Random Attractor And B(a)(ω) Is Th...mentioning

confidence: 89%