Global organization of phase space in the transition to chaos in the Lorenz system

The transition to chaos in the Lorenz system-from simple via preturbulent to chaotic dynamics-has been characterized in terms of the dynamics and the respective attractors, as described by the one-dimensional Lorenz map. In this paper we consider how this transition manifests itself globally, that is, we determine the associated organization of the entire phase space. To this end, we study how global invariant manifolds of equilibria and periodic orbits change with the parameters; the main object of study in this context is the two-dimensional (2D) stable manifold of the origin, or Lorenz manifold. We compute two-dimensional global manifolds and their complicated intersection sets with a sphere by using a boundary value problem setup. This allows us to determine how basins of attraction change or are created, and to give a precise characterization of the observed topological and geometric properties of the relevant 2D invariant manifolds during the transition.