\(\alpha\)-flips in the Lorenz system
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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