Finding First Foliation Tangencies in the Lorenz System

Abstract

Abstract. Classical studies of chaos in the well-known Lorenz system are based on reduction to the onedimensional Lorenz map, which captures the full behavior of the dynamics of the chaotic Lorenz attractor. This reduction requires that the stable and unstable foliations in a particular Poincaré section are transverse locally near the chaotic Lorenz attractor. We study when this so-called foliation condition fails for the first time and the classic Lorenz attractor becomes a quasi-attractor. This transition is characterized by the creation of tangencies between the stable and unstable foliations and the appearance of hooked horseshoes in the Poincaré return map. We consider how the threedimensional phase space is organized by the global invariant manifolds of saddle equilibria and saddle periodic orbits-before and after the loss of the foliation condition. We compute these global objects as families of orbit segments, which are found by setting up a suitable two-point boundary value problem (BVP). We then formulate a multi-segment BVP to find the first tangency between the stable foliation and the intersection curves in the Poincaré section of the two-dimensional unstable manifold of a periodic orbit. It is a distinct advantage of our BVP setup that we are able to detect and readily continue the locus of first foliation tangency in any plane of two parameters as part of the overall bifurcation diagram. Our computations show that the region of existence of the classic Lorenz attractor is bounded in each parameter plane. It forms a slanted (unbounded) cone in the three-parameter space with a curve of terminal-point, or T-point, bifurcations on the locus of first foliation tangency; we identify the tip of this cone as a codimension-three T-point-Hopf bifurcation point, where the curve of T-point bifurcations meets a surface of Hopf bifurcation. Moreover, we are able to find other first foliation tangencies for larger values of the parameters that are associated with additional T-point bifurcations: each tangency adds an extra twist to the central region of the quasi-attractor.

Wild pseudohyperbolic attractor in a four-dimensional Lorenz system

“…By robustness of the pseudohyperbolicity property, the system has the pseudohyperbolic attractor also for some neighbourhood of these parameter values. Numerically (non-rigorously) the region LA in the (σ, r)-parameter plane which corresponds to the existence of the pseudohyperbolic Lorenz attractor for fixed b = 8/3 was determined in [15,29]. The left boundary of ) corresponding to the existence of the pseudohyperbolic Lorenz attractor; the curves l 1 , l 2 and l 3 are described in [91], the curve l A=0 was first computed in [15] and studied in more detail in [29].…”

“…Poincaré map T of the section Π (the section z = r − 1 for the Lorenz model) for values of parameters (a) in LA, when the Lorenz attractor exists; (c) to the right of l A=0 . (b) The domain LA in the (σ, r)-parameter plane (for b = 8/3) corresponding to the existence of the pseudohyperbolic Lorenz attractor; the curves l 1 , l 2 and l 3 are described in[91], the curve l A=0 was first computed in[15] and studied in more detail in[29].…”

Global organization of phase space in the transition to chaos in the Lorenz system

“…We remark that the chaotic dynamics in the Lorenz system is no longer described by a one-dimensional map once is sufficiently large, the so-called foliation condition fails and there appear hooked horseshoes in the Poincaré return map [11,45]. As will be reported elsewhere, the boundary where the foliation condition fails can be continued in all parameters with a boundary value problem setup; for σ = 10 it lies at ≈ 31.01 [13]. Moreover, there is a plethora of other phenomena for larger values of , including windows of attracting periodic orbits and period-doubling to chaos [41,45] and, in particular, families of codimension-two T-point bifurcations [7,11,20]; see also [12] for how these can be found in a systematic way.…”

“…For above H , the Lorenz map has chaotic dynamics on its entire interval of definition. Note that the reduction to the one-dimensional Lorenz map is no longer exact when the stable foliation near the attractor is lost, which is the case for above about 31.01; see [11,13,45].…”

Entrainment Dynamics Organised by Global Manifolds in a Circadian Pacemaker Model

“…6 for the light/dark forcing giving instantaneous transitions at dawn/ dusk. This multi-segment approach is similar in spirit to that taken in [36,37]. This approach can be readily modified to include, for example, more realistic light protocols, such as variations in the quality and intensity of light similar to those studies in the green unicellular alga Ostreococcus tauri [38,39].…”