The Lorenz manifold as a collection of geodesic level sets

Abstract

We demonstrate a method to compute a two-dimensional global stable or unstable manifold of a vector field as a sequence of approximate geodesic level sets. Specifically, we compute the Lorenz manifold—the two-dimensional stable manifold of the origin of the well-known Lorenz equations—which has emerged as a test example for manifold computations. The information given by the geodesic level sets can be used to visualize and understand the geometry of the Lorenz manifold, and one such visualization can be seen as the cover illustration.