Visualization of geodesic level sets on global manifolds

If one wants to study the global dynamics of a given system, key components 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 using 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 circles. Our method allows to visualize the resulting surface by making use of the geodesic parametrization. This is a big advantage when one wants to understand the geometry of complicated two-dimensional manifolds, as is illustrated with examples in three- and four-dimensional vector fields