Two-dimensional global manifolds of vector fields

We describe an efficient algorithm for computing two-dimensional stable and unstable manifolds of three-dimensional vector fields. Larger and larger pieces of a manifold are grown until a sufficiently long piece is obtained. This allows one to study manifolds geometrically and obtain important features of dynamical behavior. For illustration, we compute the stable manifold of the origin spiralling into the Lorenz attractor, and an unstable manifold in zeta(3)-model converging to an attracting limit cycle

Crocheting the Lorenz manifold

You have probably seen a picture of the famous butterfly-shaped Lorenz attractor — on a book cover, a conference poster, a coffee mug or a friend’s T-shirt. The Lorenz attractor is the best known image of a chaotic or strange attractor. We are concerned here with its close cousin, the two-dimensional stable manifold of the origin of the Lorenz system, which we call the Lorenz manifold for short. This surface organizes the dynamics in the three-dimensional phase space of the Lorenz system. It is invariant under the flow (meaning that trajectories cannot cross it) and essentially determines how trajectories visit the two wings of the Lorenz attractor. We have been working for quite a while on the development of algorithms to compute global manifolds in vector fields and have computed the Lorenz manifold up to considerable size. Its geometry is very intriguing and we explored different ways of visualizing it on the computer [6, 9]. However, a real model of this surface was still lacking. During the Christmas break 2002/2003 Hinke was relaxing by crocheting hexagonal lace motifs when Bernd suggested: “Why don’t you crochet something useful? ” The algorithm we developed ‘grows ’ a manifold in steps. We start from a small disc in the stable eigenspace of the origin and add at each step a band of a fixed width. In other words, at any time of the calculation the computed part of the Lorenz manifold is a topological disc whose outer rim is (approximately) a level set of the geodesic distance from the origin. What we realized then and there is that the mesh generated by our algorithm can directly be interpreted as chrochet instructions! After some initial experimentation, the first model of the Lorenz manifold was 1 Osinga & Krauskopf Chrocheting the Lorenz manifold