Conley Index for Discrete Multivalued Dynamical Systems

The deenitions of isolating block, index pair, and the Conley index, together with the proof of homotopy and additivity properties of the index are generalized for discrete multivalued dynamical systems. That generalization provides a theoretical background of numerical computation used by Mischaikow and Mrozek in their computer assisted proof of chaos in the Lorenz equations, where nitely represented multivalued mappings appear as a tool for discretisation. 1. Introduction The aim of this paper is to construct the Conley index theory for discrete multival-ued dynamical systems, i.e. for iterates of multivalued maps. The Conley index is a topological invariant deened for isolated invariant sets in the theory of dynamical systems. Its original construction by C. Conley and his students (comp. 3]) concerned ows on locally compact metric spaces. Later the theory was generalized to arbitrary metric spaces 16, 2], to multivalued ows 12], and to discrete dynamical systems 15, 13, 4]. The Conley index theory is similar in spirit to the xed point index theory (comp. 5]) but, at least potentially, it has much broader applications. Apart from stationary ((xed) points, the theory provides existence theorems concerning bounded trajectories, heteroclinic connections, and recently also periodic trajectories 9] and chaos 10]. The main drawback of the theory is the fact that the analysis necessary to check assumptions of such theorems in concrete examples is often too complicated to be successfully preformed, despite the fact that numerical experiments indicate that the assumptions are satissed. This leads to the concept of computer assisted rigorous veriication of such assumptions 14]. Recently the rst complete and rigorous result on chaos in the Lorenz equations 11] was obtained this way on the basis of the ideas of the presented paper. It turns out that the theory of multivalued maps can be fruitfully used in studying single valued maps via computer assisted rigorous proofs. This is in contrast with the traditional motivation for studying multivalued maps, which comes from some problems in the control theory and the theory of diierential equations without uniqueness. Multivalued maps arise in these theories quite naturally as the t-translations generated by the associated multivalued ow. The most often used technique in the research on multivalued maps consists in carrying over theorems from the single valued case by means of a sequence of single valued maps approximating the multivalued map of interest. Thus, the theory of multivalued maps in this setting seems not to …