Theory And Applications Of The Conley Index For Discrete And Discontinuous Dynamical Systems

The Conley index, introduced by Charles C. Conley in his work in dealing with the differential equations of celestial mechanics, is a homotopy invariant defined on any isolated invariant sets. It is a generalization of the traditional Morse index. Conley index can be used to prove both the existence and the stability of the invariant sets, and also has the continuity property which makes the index unchanged by small disturbation. Assisted by the Morse decompositions, some complex dynamical properties can be characterized by the Conley index. All of the above perfect properties make the Conley index attract many attentions and cover extensive applications since its appearance. It is well known that the Conley index theory has been established in almost every field of dynamical systems. However, so far as we know, there is still no Conley index theory in discontinuous dynamical systems. In this thesis, we aim to tackle this hard problem and we give the definition of the Conley index for a kind of discontinuous dynamical systems, i.e., finite piecewise continuous maps. Other works include the study of the relation between discrete Conley index and bifurcation points, and the rigorous computer assisted proof of the chaotic property of the Ikeda map by Conley index. In details, the main contents of this thesis are as follows.We extend the definition of prime isolated invariant sets from flow to discrete dynamical systems, and show that any two prime isolated invariant sets do not intersect; we present a sufficient condition to detect bifurcation points by the Conley index. We originally use the viewpoint of categories to study those morphisms in the categories of the family of dynamical systems, which may have corresponding bifurcation points. We obtain a sufficient condition for two systems to have the same C-detectable bifurcation points. This condition is weaker than the conditon for complete topological conjugacy of two families of systems, so it is significant and useful for deducing from relatively simple systems or systems with known bifurcation points to the systems with unknown bifurcation points.We extend the graph of coding map from piecewise isometries to general finite piecewise continuous maps, by which we give the definition of the Conley index for finite piecewise continuous maps. We specifically discuss the difficulties caused by the discontinuity while defining the Conley index, and we design a series of methods to overcome these problems. The index we obtained still possess the Wazewski property, though weaker than the general Conley index of continuous maps, it is stronger than the index defined on continuous maps without compactness. This implies that our definition of index is meaningful. The index we designed is a true extension of the index of continuous maps and it can be utilized to prove the existence of invariant sets.We study the chaotic properties of the Ikeda map by tools of Interval Algorithm, Conley Index and Computational Homology. We prove that the Ikeda map under the given parameters contains minimal periodic points with periods 2 and 3. By the orbits connecting the period 2, 3 points, we construct the corresponding index pair, and prove that there exists a semi-conjugacy between some invariant set of Ikeda map and a shift of finite type. The positive entropy of the shift of finite type implies that the Ikeda map is chaotic under the given parameters.