### Konstantin Mischaikow and Marian Mrozeck

## Conley Index Theory

Three issues conspire to make the study of nonlinear evolution equations difficult:

- the required analysis is both global and nonlinear,
- the set of possible dynamic structures has not been and probably cannot be
classified, and
- these structures can be extremely sensitive to perturbations.

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 **C**^{0} 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