This is a review article presenting how results in the Conley index theory were applied to a computer-assisted proof of chaos in the Lorenz system. The article begins with a short description of the notions used in the proof and a historical comment on their introduction (Section 1). In Section 2, the classical retract theorem of Ważewski on the existence of a positive semitrajectory of a flow in a given set is presented. Section 3 contains the two main theorems in the theory of isolated invariant sets leading to the definition of the Conley index for flows with continuous time. In the next section the discrete-time Conley index (i.e. the index for flows being the iterates of maps), as it was introduced in the papers of the author, is presented. Properties of that index are given in Section 5. In order to link computer calculations based on numerical methods of differential equations with the theory of isolated invariant sets, the discrete-time Conley index for multivalued maps is used. It is described in Section 6. Some results on application of the Conley index theory to topological dynamics (due to C. McCord, K. Mischaikow, and the author) are presented in the next section. In particular, a result on the existence of chaos (in the sense of the semiconjugacy of an iterate of the map generating the flow to the full shift on two symbols) is given. Finally, in Section 8, the author presents the theorem (due to K. Mischaikow and the author) on the existence of chaos (in the above sense) in the Lorenz system (with parameter \(\sigma = 45\), \(R = 53\), and \(b = 10)\). It is described how the theorems from the previous sections lead to its proof, what difficulties arise in applications of computers in that proof, and what are the perspectives of use of the presented method in proofs of results on dynamics of other systems.
For the entire collection see [Zbl 0830.00005].