Multimedia supplement for the paper
Computing geodesic level sets on
global (un)stable manifolds
by Bernd
Krauskopf and Hinke Osinga
In this paper we present a general algorithm to compute the k-dimensional unstable manifold of an equilibrium or periodic orbit (or more general normally hyperbolic invariant manifold) of a vector field with an n-dimensional phase space, where 1 < k < n. Stable manifolds are computed by considering the flow for negative time. The key idea is to view the unstable manifold as a purely geometric object, hence disregarging the dynamics on the manifold, and compute it as a list of approximate geodesic level sets, which are (topological) (k-1)-spheres. Starting from a (k-1)-sphere in the linear eigenspace of the equilibrium or periodic orbit, the next geodesic level set is found in a local (and changing) coordinate system given by hyperplanes perpendicular to the last geodesic level set. In this setup the mesh points defining the approximation of the next geodesic level set can be found by solving a boundary value problem. By appropriately adding or removing meshpoints it is ensured that the mesh that represents the computed manifold is of a prescribed quality.
The general algorithm is presently implemented to compute two-dimensional manifolds in a phase space of arbitrary dimension. In this case the geodesic level sets are topological circles and the manifold is represented as a list of ribbons between consecutive level sets. We use color to distinguish between consecutive ribons or to indicate geodesic distance from the equilibrium or periodic orbit, and we also show how geodesic level sets change with increasing geodesic distance. This is very helpful when one wants to understand the often very complicated embeddings of two-dimensional (un)stable manifolds in phase space.
The properties and performance of our method are illustrated with several examples, including the stable manifold of the origin of the Lorenz system, a two-dimensional stable manifold in a four-dimensional phase space arising in a problem in optimal control, and a stable manifold of a periodic orbit that is a Möbius strip.
Figure 1 | Figure 2 | Figure 3 | |||||
Figure 4 | Figure 5 | Figure 6 | |||||
Figure 7 | Figures 8 | Figure 9 | |||||
Figure 10 |