α-flips and T-points in the Lorenz system

Abstract

We consider the Lorenz system near the classic parameter regime and study the phenomenon we call an α-flip. An α-flip is a transition where the onedimensional 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 α-flip. We also investigate how the α-flip takes place relative to the two-dimensional stable manifold of the origin, which shows no apparent topological change before or after the α-flip. Our approach allows for easy detection and subsequent two-parameter continuation of the first and further α-flips. We illustrate this for the first 25 α-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 α-flips must occur naturally near T-points. We find scaling relations for the α-flips and T-points that allow us to predict further such bifurcations and to improve the efficiency of our computations.

Finding First Foliation Tangencies in the Lorenz System

“…The bifurcation curves of the main homoclinic and heteroclinic bifurcations of p ± both terminate at a T-point. These bifurcation curves lie extremely close to each other in the (ρ, σ)-plane for fixed β = As was also shown in [18], each T-point in the Lorenz system is associated with a phenomenon called an α-flip bifurcation, where the α-limits of the respective branches of W s (p ± ) switch sides. The first, or principal, T-point was discovered by Petrovskaya and Yudovich [52] in 1980 and independently by Alsfen and Frøyland [4] in 1985.…”

“…We also compute the locus F 1 in the (ρ, σ)-plane for fixed β = 8 3 and find good agreement with the sketch by Bykov and Shilnikov. We find that F 1 is a smooth curve in this plane with a codimension-two terminal-point, or T-point, also known as a Bykov cycle [18,31,52]. At this T-point, the one-dimensional unstable manifold W u (0) is contained in the one-dimensional stable manifolds W s (p ± ), creating a heteroclinic connection from 0 to p ± ; moreover, the two-dimensional manifolds W u (p ± ) intersect W s (0) transversally, forming a heteroclinic connection from p ± to 0.…”

“…At this T-point, the one-dimensional unstable manifold W u (0) is contained in the one-dimensional stable manifolds W s (p ± ), creating a heteroclinic connection from 0 to p ± ; moreover, the two-dimensional manifolds W u (p ± ) intersect W s (0) transversally, forming a heteroclinic connection from p ± to 0. There are many other such T-points in the Lorenz system, corresponding to increasingly complicated windings of W u (0) around p ± before connecting to p ± ; see [18] for details. The bifurcation structure near a T-point has been studied, for example, in [40,43,59,66] but not in the context of the loss of the foliation condition.…”

“…The first, or principal, T-point was discovered by Petrovskaya and Yudovich [52] in 1980 and independently by Alsfen and Frøyland [4] in 1985. The first 25 T-points in the Lorenz system are shown in [18] in relation to the homoclinic and heteroclinic bifurcations of p ± and the α-flip bifurcation of W s (p ± ).…”

“…We compute them as intersection curves of stable and unstable manifolds of periodic orbits and equilibria for fixed σ = 10 and β = 8 3 . Our computations are based on continuation of a two-point boundary value problem, and we refer to [18,24,25,44] Transverse foliations for ρ = 28. As described in section 2, the chaotic attractor L consists of infinitely many surfaces each of which intersects Σ ρ in a one-dimensional curve that is part of one of the leaves of the unstable foliation [32].…”