Abstract
This paper is a survey article on computer assisted analysis of discrete dynamical systems. The focus is on the application of the Conley index theory. We discuss the definition and some properties of the Conley index and how to perform rigorous computations of it on a computer. We sketch the steps of our algorithm for computing the Conley index, which is a combination of computational homology theory and interval arithmetic. Finally, we propose a rigorous computational method for detecting homoclinic tangencies in a discrete dynamical system as an example of our argument. The problem of finding homoclinic tangencies is translated to that of finding connecting orbits in an associated dynamics on the projectivized tangent bundle and then the Conley index theory is used to find the connecting orbit. Applying this method, we prove the existence of homoclinic tangencies in the Henon family.
