TOWARDS A MORSE THEORY FOR RANDOM DYNAMICAL SYSTEMS

A generalization of the concepts of deterministic Morse theory to random dynamical systems is presented. Using the notions of attraction and repulsion in probability, the main building blocks of Morse theory such as attractor–repeller pairs, Morse sets, and the Morse decomposition are obtained for random dynamical systems.

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ATTRACTOR–REPELLER PAIR, MORSE DECOMPOSITION AND LYAPUNOV FUNCTION FOR RANDOM DYNAMICAL SYSTEMS

Stochastics and Dynamics ◽

10.1142/s0219493708002494 ◽

2008 ◽

Vol 08 (04) ◽

pp. 625-641 ◽

Cited By ~ 4

Author(s):

ZHENXIN LIU ◽

SHUGUAN JI ◽

MENGLONG SU

Keyword(s):

Dynamical Systems ◽

Lyapunov Function ◽

Metric Space ◽

Stability Theory ◽

Lyapunov Functions ◽

Random Dynamical Systems ◽

Morse Decomposition ◽

Compact Metric Space ◽

Morse Decompositions ◽

The Stability

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 [7,17], 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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Soluble and Non-Aggregated Phthalocyanines: Synthesis, Mechanistic Aspects and Their Main Building Blocks

Current Organic Synthesis ◽

10.2174/1570179414666170403115907 ◽

2018 ◽

Vol 14 (8) ◽

Cited By ~ 2

Author(s):

Nicholas R.S. Gobo ◽

Timothy J. Brocksom ◽

Kleber T. de Oliveira

Keyword(s):

Building Blocks ◽

Main Building

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VARIATIONAL PRINCIPLE FOR SUBADDITIVE SEQUENCE OF POTENTIALS IN BUNDLE RANDOM DYNAMICAL SYSTEMS

Stochastics and Dynamics ◽

10.1142/s0219493709002634 ◽

2009 ◽

Vol 09 (02) ◽

pp. 205-215 ◽

Cited By ~ 2

Author(s):

XIANFENG MA ◽

ERCAI CHEN

Keyword(s):

Dynamical Systems ◽

Variational Principle ◽

Multifractal Analysis ◽

Weak Condition ◽

Random Dynamical Systems ◽

Topological Pressure ◽

Set Up

The topological pressure is defined for subadditive sequence of potentials in bundle random dynamical systems. A variational principle for the topological pressure is set up in a very weak condition. The result may have some applications in the study of multifractal analysis for random version of nonconformal dynamical systems.

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