Simplicial Models for the Global Dynamics of Attractors

Abstract

Given an unknown attractor A in a continuous dynamical system, how we can discover the topology and dynamics of A? As a practical matter, how can we do so from only a finite amount of information? One way of doing so is to produce a semi-conjugacy from A onto a model system M whose topology and dynamics are known. The complexity of M then provides a lower bound for the complexity of A. In this paper, we use the techniques of the Conley index to construct a simplicial model and a surjective semi-conjugacy for a large class of attractors. The essential features of this construction are that the model M can be explicitly described and that the finite amount of information needed to construct it is computable.