A Conley index study of the evolution of the Lorenz strange set

Abstract

In this paper we study the Lorenz equations using the perspective of the Conley index theory. More specifically, we examine the evolution of the strange set that these equations posses throughout the different values of the parameter. We also analyze some natural Morse decompositions of the global attractor of the system and the role of the strange set in these decompositions. We calculate the corresponding Morse equations and study their change along the successive bifurcations. We see how the main features of the evolution of the Lorenz system are explained by properties of the dynamics of the global attractor. In addition, we formulate and prove some theorems which are applicable in more general situations. These theorems refer to Poincaré–Andronov–Hopf bifurcations of arbitrary codimension, bifurcations with two homoclinic loops and a study of the role of the traveling repellers in the transformation of repeller–attractor pairs into attractor–repeller ones.