Summary
If one wants to study the global dynamics of a given system, key com ponents are the stable or unstable manifolds of invariant sets, such as equilibria and periodic orbits. Even in the simplest examples, these global manifolds must be approximated by means of numerical computations. We discuss an algorithm for computing global manifolds of vector fields that is decidedly geometric in nature. A two-dimensional manifold is built up as a collection of approximate geodesic level sets, i.e. topological smooth circles. Our method allows to visualize the resulting surface by making use of the geodesic parametrization.
As we show with the example of the Lorenz system, this is a big advantage when one wants to understand the geometry of complicated two-dimensional global man ifolds. More precisely, for the standard system parameters, the origin of the Lorenz system has a two-dimensional stable manifold — called the Lorenz manifold — and the other two equilibria each have a two-dimensional unstable manifold. The inter sections 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.
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Krauskopf, B., Osinga, H.M., Doedel, E.J. (2009). Visualizing global manifolds during the transition to chaos in the Lorenz system. In: Hege, HC., Polthier, K., Scheuermann, G. (eds) Topology-Based Methods in Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88606-8_9
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DOI: https://doi.org/10.1007/978-3-540-88606-8_9
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