Finding first foliation tangencies in the Lorenz system

This is the final version of the article. Available from SIAM via the DOI in this record.Classical studies of chaos in the well-known Lorenz system are based on reduction to the one-dimensional Lorenz map, which captures the full behavior of the dynamics of the chaotic Lorenz attractor. This reduction requires that the stable and unstable foliations on a particular Poincar e section are transverse locally near the chaotic Lorenz attractor. We study when this so-called foliation condition fails for the rst 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 Poincar e return map. We consider how the three-dimensional 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 nd the rst tangency between the stable foliation and the intersection curves in the Poincar e section of the two-dimensional unstable manifold of a periodic orbit. It is a distinct advantage of our BVP set-up that we are able to detect and readily continue the locus of rst 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 rst 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 nd other rst 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

Parameter exclusions in Henon-like systems

This survey is a presentation of the arguments in the proof that Henon-like maps f_a(x,y)=(1-a x^2,0) + R(a,x,y) with |R(a,x,y)|< b have a "strange attractor", with positive Lebesgue probability in the parameter "a", if the perturbation size "b" is small enough. We first sketch a "geometric model" of the strange attractor in this context, emphasising some of its key geometrical properties, and then focus on the construction and estimates required to show that this geometric model does indeed occur for many parameter values. Our ambitious aim is to provide an exposition at one and the same time intuitive, synthetic, and rigorous. We think of this text as an introduction and study guide to the original papers in which the results were first proved. We shall concentrate on describing in detail the overall structure of the argument and the way it breaks down into its (numerous) constituent sub-arguments, while referring the reader to the original sources for detailed technical arguments.Comment: 40 pages, 3 figure

Chaotic dynamics of three-dimensional H\'enon maps that originate from a homoclinic bifurcation

We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers (\lambda e^{i\vphi}, \lambda e^{-i\vphi}, \gamma), where $0<\lambda<1<|\gamma|$ and $|\lambda^2\gamma|=1$. We show that in a three-parameter family, g_{\eps}, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near \eps =0 where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional H\'enon-like map. This map possesses, in some parameter regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional H\'enon maps occupy in the class of quadratic volume-preserving diffeomorphisms.Comment: laTeX, 25 pages, 6 eps figure

Global surfaces of section for Reeb flows in dimension three and beyond

We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are also presented.Comment: 33 pages, 3 figures. This is an extended version of a paper written for Proceedings of the ICM, Rio 2018; in v3 we made minor additional corrections, updated references, added a reference to work of Lu on the Conley Conjectur

Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial

A new computational technique based on the symbolic description utilizing kneading invariants is proposed and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique allows for uncovering the stunning complexity and universality of bi-parametric structures and detect their organizing centers - codimension-two T-points and separating saddles in the kneading-based scans of the iconic Lorenz equation from hydrodynamics, a normal model from mathematics, and a laser model from nonlinear optics.Comment: Journal of Bifurcations and Chaos, 201