An Algorithmic Approach to Chain Recurrence

Abstract

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 Morse sets 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.KEYWORDS. chain recurrence, Lyapunov function, Conley's decomposition theorem, algorithms, computation. AMS SUBJECT CLASSIFICATION. 37M99, 37C70.Conley's Fundamental Decomposition Theorem and its extension to Morse decompositions is a powerful tool in dynamical systems theory. However, the framework on which the standard theory is built is not does not lead naturally to an algorithmic or computational approach for the approximation of the chain recurrent set, i.e. generation of Morse decompositions or the approximation of a Lyapunov function for the gradient-like part of the system. One can approximate the chain recurrent set by the -chain recurrent set for finite > 0, but there are no algorithmic or computational techniques for computing this set directly.In this paper, we present an alternative approach based on finite discretizations and combinatorial multivalued maps. This approach has several advantages. The basic elements of the theory can be proved in a straightforward manner. Moreover, the methods are inherently combinatorial and hence algorithmic. The framework leads naturally to computational techniques for analyzing qualitative dynamics including rigorous computer-assisted proofs, see e.g. [10, 14, 4, 5].

Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions

“…In (Kalies et al, 2005;Ban and Kalies, 2006;Goullet et al, 2015) a computational approach to construct complete Lyapunov functions was considered. A discrete-time system was given by the time-T map and the phase space was subdivided into cells and the dynamics between them computed through an induced multivalued map using the computer package GAIO (Dellnitz et al, 2001).…”

Revised CPA method to compute Lyapunov functions for nonlinear systems

“…Norton [26] even suggested that this characterization should be referred to as the Fundamental Theorem of Dynamical Systems. In [21], Kalies, Mischaikow and VanderVorst present an algorithmic approach to construct approximations to complete Lyapunov functions for discrete dynamical systems. By considering the time-T map of a continuous system, this method can be used to find an approximation to a complete Lyapunov function for a con-tinuous dynamical system as well.…”

Lattice structures for attractors I

“…This theorem implies that there is no fundamental algebraic obstruction to identifying any finite collection of attractors via attracting neighborhoods, which are computable objects [10]. Another consequence is an alternative proof of the existence of a index filtration to that presented in [8].…”

“…As is demonstrated in [10], given a finite discretization of X and finite approximation of ϕ, elements of ANbhd(ϕ) and RNbhd(ϕ) can be computed. Part of the motivation for this paper is our belief that useful extensions and proper analysis of these types of computational methods requires a greater understanding of the above mentioned algebraic structures.…”