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

2008 ◽  
Vol 08 (04) ◽  
pp. 625-641 ◽  
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.

Strong Morse–Lyapunov functions for Morse decompositions of attractors of random dynamical systems

2017 ◽  
Vol 18 (01) ◽  
pp. 1850012
Author(s):  
Xuewei Ju ◽  
Ailing Qi ◽  
Jintao Wang
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Functions ◽  
Random Dynamical Systems ◽  
Random Attractors ◽  
Morse Decompositions ◽  
The Stability

In this paper, we first construct strong Lyapunov functions for random attractors. On this basis, we then establish strong Morse–Lyapunov functions for Morse decompositions of the random attractors. Finally, the stability of the Morse decompositions is studied through the Morse–Lyapunov functions.

Topological techniques for efficient rigorous computation in dynamics

Acta Numerica ◽  
2002 ◽  
Vol 11 ◽  
pp. 435-477 ◽  
Author(s):  
Konstantin Mischaikow
Keyword(s):  
Dynamical Systems ◽  
Decomposition Theorem ◽  
Topological Methods ◽  
Infinite Dimensional ◽  
Isolating Blocks ◽  
Morse Decompositions ◽  
Rigorous Computation

We describe topological methods for the efficient, rigorous computation of dynamical systems. In particular, we indicate how Conley's Fundamental Decomposition Theorem is naturally related to combinatorial approximations of dynamical systems. Furthermore, we show that computations of Morse decompositions and isolating blocks can be performed efficiently. We conclude with examples indicating how these ideas can be applied to finite- and infinite-dimensional discrete and continuous dynamical systems.

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