Visualizing the transition to chaos in the Lorenz system

HM Osinga; B Krauskopf; E.J. Doedel

Visualizing the transition to chaos in the Lorenz system

HM Osinga, B Krauskopf, E.J. Doedel

Department of Engineering Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference Contribution (Conference Proceeding)

Abstract

The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. This paper addresses the role of the global stable and unstable manifolds in organising the dynamics. More precisely, for the standard system parameters, the origin has a two-dimensional stable manifold and the other two equilibria each have a two-dimensional unstable manifold. The intersections of these manifolds in the three-dimensional phase space form heteroclinic connections from the nontrivial equilibria to the origin. A parameter-dependent visualization of these manifolds clarifies the transition to chaos in the Lorenz system

Translated title of the contribution	Visualizing the transition to chaos in the Lorenz system
Original language	English
Title of host publication	Topology-Based Methods in Visualization 2007
Publication status	Published - Nov 2006

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Name and Venue of Event: Grimma, Germany

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http://rose.bris.ac.uk/dspace/handle/1983/879

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Visualizing the transition to chaos in the Lorenz system
Osinga, H., Krauskopf, B. & Doedel, E., Nov 2006.
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@inproceedings{6e5d6aed7c5146a9822619d03cc748bc,

title = "Visualizing the transition to chaos in the Lorenz system",

abstract = "The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. This paper addresses the role of the global stable and unstable manifolds in organising the dynamics. More precisely, for the standard system parameters, the origin has a two-dimensional stable manifold and the other two equilibria each have a two-dimensional unstable manifold. The intersections of these manifolds in the three-dimensional phase space form heteroclinic connections from the nontrivial equilibria to the origin. A parameter-dependent visualization of these manifolds clarifies the transition to chaos in the Lorenz system",

author = "HM Osinga and B Krauskopf and E.J. Doedel",

note = "Name and Venue of Event: Grimma, Germany",

year = "2006",

month = nov,

language = "English",

booktitle = "Topology-Based Methods in Visualization 2007",

}

TY - GEN

T1 - Visualizing the transition to chaos in the Lorenz system

AU - Osinga, HM

AU - Krauskopf, B

AU - Doedel, E.J.

N1 - Name and Venue of Event: Grimma, Germany

PY - 2006/11

Y1 - 2006/11

N2 - The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. This paper addresses the role of the global stable and unstable manifolds in organising the dynamics. More precisely, for the standard system parameters, the origin has a two-dimensional stable manifold and the other two equilibria each have a two-dimensional unstable manifold. The intersections of these manifolds in the three-dimensional phase space form heteroclinic connections from the nontrivial equilibria to the origin. A parameter-dependent visualization of these manifolds clarifies the transition to chaos in the Lorenz system

AB - The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. This paper addresses the role of the global stable and unstable manifolds in organising the dynamics. More precisely, for the standard system parameters, the origin has a two-dimensional stable manifold and the other two equilibria each have a two-dimensional unstable manifold. The intersections of these manifolds in the three-dimensional phase space form heteroclinic connections from the nontrivial equilibria to the origin. A parameter-dependent visualization of these manifolds clarifies the transition to chaos in the Lorenz system

M3 - Conference Contribution (Conference Proceeding)

BT - Topology-Based Methods in Visualization 2007

ER -

Visualizing the transition to chaos in the Lorenz system

Abstract

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Research output

Visualizing the transition to chaos in the Lorenz system

Cite this