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].

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ABSTRACT. We describe the basic lattice structures of attractors and repellers in dynamical systems. The structure of distributive lattices allows for an algebraic treatment of gradient-like dynamics in general dynamical systems, both invertible and noninvertible. We separate those properties which rely solely on algebraic structures from those that require some topological arguments, in order to lay a foundation for the development of algorithms to manipulate these structures computationally.

ABSTRACT. The algebraic structure of the attractors in a dynamical system determine much of its global dynamics. The collection of all attractors has a natural lattice structure, and this structure can be detected through attracting neighborhoods, which can in principle be computed. Indeed, there has been much recent work on developing and implementing general computational algorithms for global dynamics, which are capable of computing attracting neighborhoods efficiently. Here we address the question of whether all of the algebraic structure of attractors can be captured by these methods.

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