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.
We study $C^{d,r}$-Hénon-like families $(f_{a\, b})_{a\, b}$ with two parameters $(a,b)\in \mathbb R^2$. We show the existence of an open set of parameters $(a,b)\in \mathcal D$, so that a renormalization chart conjugates an iterate of $f_{a\, b}$ to a perturbation of $(x,y)\mapsto ((x^2+c_1)^2+c_2,0)$. We prove that the map $(a,b)\in \mathcal D\mapsto (c_1,c_2)$ is a $C^d$-diffeomorphism; as first numerically conjectured by Milnor in 1992. Furthermore, we show the existence of an open set of parameters $(a,b)$ so that $f_{a\, b}$ displays exactly two different renormalized Hénon-like maps who…
We numerically study quasiperiodic normally hyperbolic attracting invariant circles that appear for certain parameter values in a family of three-dimensional Henon-like maps. These parameter values make up contour segments in the parameter space where the invariant circles have constant irrational rotation number. At the edges of these segments we find a breakdown of the hyperbolicity of the invariant circle. We observe the collision and loss of smoothness of two of the invariant Lyapunov bundles while the Lyapunov exponents all remain distinct. This is very similar to the breakdown of quasipe…
Homoclinic tangencies to resonant saddles and discrete Lorenz attractors
研究了一类二次同宿切线到鞍不动点的三参数一般开折中周期轨道的分支。我们应用重新标度技术的第一次返回（庞加莱）的地图，并表明，重新标度的地图可以带来一个地图渐近接近的三维Henon地图$\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$，如已知的，表现出野生双曲洛伦兹吸引子在一些开放的参数域。在此基础上，我们证明了类Lorenz吸引子的无穷级联的存在性。
We study geometrical and dynamical properties of the so-called discrete Lorenz-like attractors, that can be observed in three-dimensional diffeomorphisms. We propose new phenomenological scenarios of their appearance in one parameter families of such maps. We pay especially our attention to such a scenario that can lead to period-2 Lorenz-like attractors. These attractors have very interesting dynamical properties and we show that their crises can lead, in turn, to the emergence of pseudohyperbolic discrete Lorenz shape attractors of new types. We also show examples of all these attractors in …
We prove that the coexistence of infinitely many prevalent Hénon-like phenomena is Kolmogorov typical in sectional dissipative $C^{d,r}$-Berger domains of parameter families of diffeomorphisms of dimension $m\geq 3$ for $d<r-1$. Namely, we answer an old question posed by Colli in [Annales de l'Institut Henri Poincare-Nonlinear Analysis, 15, 539--580 (1998)] on typicality of the coexistence of infinitely many non-hyperbolic strange attractors for $3\leq d<r-1$.
We present a comprehensive mechanism for the emergence of rotational horseshoes and strange attractors in a class of two-parameter families of periodically-perturbed differential equations defining a flow on a three-dimensional manifold. When both parameters are zero, its flow exhibits an attracting heteroclinic network associated to two periodic solutions. After slightly increasing both parameters, while keeping a two-dimensional connection unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the periodic solutions do not intersect. We prove a wide ra…
