Three issues conspire to make the study of nonlinear evolution equations difficult:
With this in mind Charles Conley focussed attention on isolating neighborhoods and developed an index (now known as the Conley index) of isolating neighborhoods. Isolating neighborhoods are stable with respect to C0 perturbations as is the index. Furthermore, the index can be used to obtain information concerning the structure of the associated isolated invariant set.
In these notes we will describe isolating neighborhoods, their associated isolated invariant sets, and decompositons of these sets, e.g. Morse decompositions. We will discuss the Conley index and associated theorems which allow one to use the index to conclude the existence of some of the most fundamental dynamic structures such as: fixed points, periodic orbits, connecting orbits, and horseshoes. We will also indicate how these results have been used in the context of differential equations.
With these theoretical tools in place we will turn to the question of computing the index with special emphasis on numerical techniques. To do this we will discuss the Conley theory in the setting of multivalued maps. The multivaluedness is used to control the errors inherent in any numerical method. To indicated the power of this approach we will describe several computer assisted proofs of the existence of chaotic dynamics. Since these methods are rather new, there are many potential directions for further development on which we shall speculate.
Konstantin Mischaikow and Marian Mrozeck
Jul 14 1998