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The Lorenz System Near the Loss of the Foliation Condition

The well-known three-dimensional Lorenz system is classically studied via its reduction to the one-dimensional Lorenz map, which captures the full behaviour of the dynamics of the system. The reduction requires the existence of a stable foliation. We study a parameter regime where this so-called foliation condition fails for the first time and, consequently, the Lorenz map no longer accurately represents the dynamics of the full system. To this end, we study how the three-dimensional phase space of the Lorenz system is organised by the global invariant manifolds of saddle equilibria and saddle periodic orbits. Specifically, we explain and define two phenomena, observed by Sparrow in the 1980’s. First, the so-called ‘half-swing’ of the one-dimensional stable manifolds W s(p±) of the secondary equilibria p± from one side to the other. Secondly, the development of hooks in the Poincaré return map that marks the loss of the foliation condition. To investigate both these phenomena, we make extensive use of the continuation of orbit segments formulated by two-point boundary value problems. To explain the ‘half-swing’ we characterise geometrically a bifurcation in the α-limit of W s(p±), which we call an α-flip. We accurately compute the parameter value at which this first α-flip occurs and find many subsequent α-flips. In two parameters, we show that each α-flip ends at a different codimension-two bifurcation point, called a T-point, many of which have not been found before. In particular, we discuss the α-flip in the context of the known bifurcation structure around the first T-point. To study the foliation condition we calculate the intersection curves of the two-dimensional unstable manifold W (Γ) of a periodic orbit Γ with the classic Poincaré section. We identify when hooks form in the Poincaré map as a point of tangency of W (Γ) with the stable foliation. Subsequent continuations show that this point lies on a curve through the first T-point in two-parameter space, which provides a connection with the α-flip. Our approach allows us to identify in a convenient way the region of existence of the classic Lorenz attractor in twoand three-parameter space.