$$\alpha$$-flips in the Lorenz system

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