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

Héctor Barge, J. M. R. Sanjurjo

TL;DR

The paper reframes the Lorenz system through Conley index theory to illuminate how the strange set $K_r$ evolves with the parameter $r$ and how Morse decompositions restructure the global attractor. It establishes a sequence of bifurcation-driven topological changes, including generalized pitchfork/Hopf scenarios and a transient-chaos regime, by computing Conley indices and Morse equations for relevant invariant sets and their decompositions. A key result is that the strange set has the cohomology of a circle ($CH^*(K_r)\cong H^*(S^1,*)$) during preturbulence and transitions to a figure-eight cohomology, while a traveling repeller mechanism drives the creation of the Lorenz attractor at $r=24.06$ and its subsequent Hopf bifurcation at $r=24.74$, with Morse equations capturing each stage. The work provides a general, topological framework for tracking parameter-induced transitions in chaotic flows, with a travelling-repeller theorem linking repeller–attractor and attractor–repeller decompositions that is applicable beyond the Lorenz system.

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 Poincaré-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.

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

TL;DR

The paper reframes the Lorenz system through Conley index theory to illuminate how the strange set evolves with the parameter and how Morse decompositions restructure the global attractor. It establishes a sequence of bifurcation-driven topological changes, including generalized pitchfork/Hopf scenarios and a transient-chaos regime, by computing Conley indices and Morse equations for relevant invariant sets and their decompositions. A key result is that the strange set has the cohomology of a circle () during preturbulence and transitions to a figure-eight cohomology, while a traveling repeller mechanism drives the creation of the Lorenz attractor at and its subsequent Hopf bifurcation at , with Morse equations capturing each stage. The work provides a general, topological framework for tracking parameter-induced transitions in chaotic flows, with a travelling-repeller theorem linking repeller–attractor and attractor–repeller decompositions that is applicable beyond the Lorenz system.

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 Poincaré-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.

Paper Structure

This paper contains 5 sections, 5 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Pitchfork bifurcations
  4. Transient chaos
  5. Travelling repellers: the creation and evolution of the Lorenz attractor

Key Result

Theorem 1

Let $\varphi _{\lambda }:\mathbb{R}^{n}\times \mathbb{R}\rightarrow \mathbb{R}^{n}$ be a family of flows induced by a system $\dot{X}=F_{\lambda }(X)$ of ODE in $\mathbb{R}^{n}$ depending differentiably on a parameter $\lambda \in \lbrack 0,1]$ and suppose that $0$ is an equilibrium for every $\lamb The family $(K_{\lambda })$ also shrinks to $0$. In the particular case of the Lorenz flow, $A_{\la

Figures (1)

  • Figure 1: Branched manifold

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Lemma 6
  • proof
  • ...and 1 more