Global organization of phase space in the transition to chaos in the Lorenz system
Eusebius J. Doedel, Bernd Krauskopf, and Hinke M. Osinga
Abstract
The transition to chaos in the Lorenz system — from simple via preturbulent to chaotic dynamics — has been characterized in terms of the dynamics on 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 stable manifold of the origin, or Lorenz manifold. Two-dimensional global manifolds and their complicated intersection sets with a plane or sphere are calculated with 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 invariant manifolds during the transition. In particular, we show where preturbulence occurs after the first homoclinic bifurcation and how it suddenly disappears in a heteroclinic bifurcation to give rise to a chaotic attractor and its basin.
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