Global bifurcations of the Lorenz manifold

EJ Doedel, B Krauskopf, HM Osinga

Research output: Contribution to journalArticle (Academic Journal)peer-review

58 Citations (Scopus)


In this paper we consider the interaction of the Lorenz manifold - the two-dimensional stable manifold of the origin of the Lorenz equations - with the two-dimensional unstable manifolds of the secondary equilibria or bifurcating periodic orbits of saddle type. We compute these manifolds for varying values of the parameter rho in the Lorenz equations, which corresponds to the transition from simple to chaotic dynamics with the classic Lorenz butterfly attractor at rho = 28. Furthermore, we find and continue in rho the first 512 generic heteroclinic orbits that are given as the intersection curves of these two- dimensional manifolds. The branch of each heteroclinic orbit emerges from the well- known first codimension-one homoclinic explosion point at rho approximate to 13.9265, has a fold and then ends at another homoclinic explosion point with a specific rho-value. We describe the combinatorial structure of which heteroclinic orbit ends at which homoclinic explosion point. This is verified with our data for the 512 branches from which we automatically extract (by means of a small computer program) the relevant symbolic information. Our results on the manifold structure are complementary to previous work on the symbolic dynamics of periodic orbits in the Lorenz attractor. We point out the connections and discuss directions for future research.
Translated title of the contributionGlobal bifurcations of the Lorenz manifold
Original languageEnglish
Pages (from-to)2947 - 2972
Number of pages26
Volume19 (12)
Publication statusPublished - Dec 2006

Bibliographical note

Publisher: IOP Publishing Ltd
Other: with multimedia supplement


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