An Algorithmic Approach to Chain Recurrence

Authors:
W. D. Kalies;K. Mischaikow;R. C. A. M. VanderVorst
Affiliations:
Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33434, USA;Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, GA 30332, USA;Department of Mathematics, Vrije University Amsterdam, De Boelelaan 1081, 1081 HV, Amsterdam, The Netherlands
Venue:
Foundations of Computational Mathematics
Year:
2005

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Abstract

In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. This approach allows for an algorithmic construction of isolating blocks for the components of Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley’s Lyapunov function. This is a natural framework in which to develop computational techniques for the analysis of qualitative dynamics including rigorous computer-assisted proofs.