Visualizing Lorenz Manifold

This paper begins by providing a mathematical characterization of the Lorenz system. Then, we present a 1D algorithm for computing the global one-dimensional unstable manifold of a saddle point of Lorenz system. However, a higher-dimensional stable and unstable manifold of Lorenz system consist of infinitely many orbits, and a finite collection of orbits typically does not give an acceptable picture of the Lorenz system. We try to find a sophisticated algorithm for computing two-dimensional stable and unstable Lorenz manifolds of three-dimensional vector field. We are concerned with the technically challenging case of computing stable and unstable manifolds of Lorenz system in the three-dimensional vector fields in this paper.