Jennifer L. Creaser, Bernd Krauskopf, and Hinke M. Osinga

**Abstract**

We consider the Lorenz system near the classic parameter regime and study the phenomenon we call an \(\alpha\)-flip. An \(\alpha\)-flip is a transition where the one-dimensional stable manifolds \(W^{s}(p^{\pm})\) of two secondary equilibria \(p^{\pm}\) undergo a sudden transition in terms of the direction from which they approach \(p^{\pm}\). This fact was discovered by Sparrow in the 1980's but the stages of the transition could not be calculated and the phenomenon was not well understood [C. Sparrow, *The Lorenz equations*, Springer-Verlag New York, 1982]. Here we employ a boundary value problem set-up and use pseudo-arclength continuation in AUTO> to follow this sudden transition of \(W^{s}(p^{\pm})\) as a continuous family of orbit segments. In this way, we geometrically characterize and determine the moment of the actual \(\alpha\)-flip. We also investigate how the \(\alpha\)-flip takes place relative to the two-dimensional stable manifold of the origin, which shows no apparent topological change before or after the \(\alpha\)-flip. Our approach allows for easy detection and subsequent two-parameter continuation of the first and further \(\alpha\)-flips. We illustrate this for the first 25 \(\alpha\)-flips and find that they end at terminal points, or T-points, where there is a heteroclinic connection from the secondary equilibria to the origin. We find scaling relations for the \(\alpha\)-flips and T-points that allow us to predict further such bifurcations and to improve the efficiency of our calculations.

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Created by Hinke Osinga

Last modified: Tue Feb 25 17:37:43 NZDT 2014