A Computational Approach to Conley’s Decomposition Theorem

[+] Author and Article Information
Hyunju Ban

Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33434hban@fau.edu

William D. Kalies

Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33434wkalies@fau.edu

J. Comput. Nonlinear Dynam 1(4), 312-319 (May 14, 2006) (8 pages) doi:10.1115/1.2338651 History: Received November 15, 2005; Revised May 14, 2006

Background. The discrete dynamics generated by a continuous map can be represented combinatorially by an appropriate multivalued map on a discretization of the phase space such as a cubical grid or triangulation. Method of approach. We describe explicit algorithms for computing dynamical structures for the combinatorial multivalued maps. Results. We provide computational complexity bounds and numerical examples. Specifically we focus on the computation attractor-repeller pairs and Lyapunov functions for Morse decompositions. Conclusions. The computed discrete Lyapunov functions are weak Lyapunov functions and well-approximate a continuous Lyapunov function for the underlying map.

Copyright © 2006 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Directed graph representation of the minimal multivalued map for the logistic map f(x)=2.5x(1−x). The pair ({G5},{G1}) is an attractor-repeller pair for F.

Grahic Jump Location
Figure 2

Lyapunov function for logistic map for (a)λ=3.3 and n=217=131,072 and (b)λ=3.385 and n=220=1,048,576

Grahic Jump Location
Figure 3

Lyapunov function for the van der Pol ODE. In (b) the grayscale represents the height.



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