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

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

Eusebius J. Doedel, Bernd Krauskopf & Hinke M. Osinga

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 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 two-dimensional invariant manifolds during the transition.

Animation with Figure 3: Intersection of the Lorenz manifold for \(\varrho = 10.0\) with the sphere \(S_R\) for \(R = 67.156\).

Animation with Figure 4: Intersection of the Lorenz manifold for \(\varrho = 18.0\) with the sphere \(S_R\) for \(R = 69.062\).

Animation with Figure 10: Intersection of the Lorenz manifold for \(\varrho = 28.0\) with the sphere \(S_R\) for \(R = 70.709\).

Animation with Figure 11: Data from the single continuation run to compute the intersection of the Lorenz manifold for \(\varrho = 23.0\) with the sphere \(S_R\) for \(R = 69.947\).

Animation with Figure 14: One isola, where the Lorenz manifold for \(\varrho = 28.0\) intersects \(S_R\) with \(R = 70.709\) from the outside coming back in, shown together with the primary intersection curves.

AUTO demos: The Python drivers for all AUTO calculations are available as a gzipped tar file Lorenz.tar.gz. Download and unpack by typing

tar xvfz Lorenz.tar.gz

	Animation with Figure 3: Intersection of the Lorenz manifold for \(\varrho = 10.0\) with the sphere \(S_R\) for \(R = 67.156\).
	Animation with Figure 4: Intersection of the Lorenz manifold for \(\varrho = 18.0\) with the sphere \(S_R\) for \(R = 69.062\).
	Animation with Figure 10: Intersection of the Lorenz manifold for \(\varrho = 28.0\) with the sphere \(S_R\) for \(R = 70.709\).
	Animation with Figure 11: Data from the single continuation run to compute the intersection of the Lorenz manifold for \(\varrho = 23.0\) with the sphere \(S_R\) for \(R = 69.947\).
	Animation with Figure 14: One isola, where the Lorenz manifold for \(\varrho = 28.0\) intersects \(S_R\) with \(R = 70.709\) from the outside coming back in, shown together with the primary intersection curves.
	AUTO demos: The Python drivers for all AUTO calculations are available as a gzipped tar file `Lorenz.tar.gz`. Download and unpack by typing `tar xvfz Lorenz.tar.gz`