Isolating Segments, Fixed Point Index, and Symbolic Dynamics: III. Applications

Conley Index of Poincaré Maps in Isolating Segments

We calculate the discrete-time Conley index of the Poincaré map of a time-periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment. As an application, we present results on the existence of bounded solutions of some planar equations.

Fixed Point Theorems and Applications

Symbolic Dynamics and Coding Applications

This article is a summary of the plenary lecture that I gave at the 1995 IEEE International Symposium on Information Theory. The lecture was intended to give a concrete and elementary introduction to the subject of symbolic dynamics. From my point of view, symbolic dynamics is the subject of \coding" from one \constrained set" of sequences to another. Of course, there are restrictions on what I...

Retracts, fixed point index and differential equations

Some problems in differential equations evolve towards Topology from an analytical origin. Two such problems will be discussed: the existence of solutions asymptotic to the equilibrium and the stability of closed orbits of Hamiltonian systems. The theory of retracts and the fixed point index have become useful tools in the study of these questions. Retractos, ı́ndice de punto fijo y ecuaciones d...

The Conley Index and Symbolic Dynamics

Suppose that N is an isolating neighborhood for S with respect to a continuous map f which decomposes into a disjoint union of nitely many compact sets. We establish a relationship between the Conley indices of subsets of S which appear in a natural way when studying the behavior of f from the viewpoint of symbolic dynamics. The relationship is then used to prove a simple criterion for chaos in...