α-flips and T-points in the Lorenz system

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(+/-)) of two secondary equilibria p(+/-) undergo a sudden transition in terms of the direction from which they approach p(+/-). This is a bifurcation at infinity and does not involve an invariant object in phase space. This fact was discovered by Sparrow in the 1980s but the stages of the transition could not be calculated and the phenomenon was not well understood (Sparrow 1982 The Lorenz equations (New York: Springer)). 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(+/-)) 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. It turns out that alpha-flips must occur naturally near T-points. 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 computations.