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Globalizing two-dimensional unstable manifolds of maps

Bernd Krauskopf
Theoretical Physics
Free University
De Boelelaan 1081
1081 HV Amsterdam
The Netherlands

Hinke Osinga
The Geometry Center
University of Minnesota
1300 South Second Street
Minneapolis, MN 55454


We present an algorithm for computing the global two-dimensional unstable manifold of a normally hyperbolic invariant circle of a three-dimensional map. Our algorithm computes intersections of the unstable manifold with a finite number of leaves of a chosen linear foliation. This allows us to guarantee the quality of the mesh on this manifold. We compute growing pieces of the unstable manifold by using a method that does not depend on the dynamics on the manifold. Our method can also be used for the computation of global two-dimensional unstable manifolds of hyperbolic saddle points. The global stable manifold is obtained by considering the inverse map.

The map in question may be defined explicitly or as the Poincaré map of a vector field. The latter allows us to compute the two-dimensional unstable manifold that corresponds to a three-dimensional manifold of a normally hyperbolic torus or a limit cycle in dimension four. The performance of our algorithm is demonstrated with examples of the different cases. The choices for the foliation are discussed.

To appear in Int. J. Bifurcation & Chaos


1. Introduction
2. Overview of earlier methods
3. The algorithm

3.1 Iterating a fundamental domain
3.2 Globalization by adding discrete circles
3.3 The case of a hyperbolic fixed point
3.4 Mesh adaptation

4. Correctness
5. Examples

5.1 The 3D-fattened Arnol'd family
5.2 Quasiperiodically forced Hénon map
5.3 A saddle surface
5.4 The Lorenz system
5.5 Normal form of the Hopf-Hopf bifurcation

6. Discussion

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Written by: Bernd Krauskopf & Hinke Osinga
Comments to: hinke@geom.umn.edu
Created: May 8 1997 --- Last modified: Thu Sep 11 17:02:35 1997