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

## TL;DR

This paper applies Conley index theory to analyze the evolution of the Lorenz system's strange set across parameter changes, providing new insights into bifurcations and attractor decomposition.

## Contribution

It introduces a Conley index framework for studying Lorenz attractor evolution and proves theorems applicable to complex bifurcations.

## Key findings

- Morse equations change along bifurcations
- Theorems on bifurcations with homoclinic loops
- Role of traveling repellers in attractor transformations

## 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. In addition, we formulate and prove some theorems which are applicable in more general situations. These theorems refer to Poincar\'{e}-Andronov-Hopf bifurcations of arbitrary codimension, bifurcations with two homoclinic loops and a study of the role of the travelling repellers in the transformation of repeller-attractor pairs into attractor-repeller ones.