Constructive trapping domains and Smale-horseshoe conditions for the multidimensional Hénon map with general nonlinearity

The (n+1)-dimensional Hénon-like map $\bar{x}=f(x)+\mathbb{1}y$, $\bar{y}=\mathbf{b}x+Ay$ (with $A$ a lower-shift matrix) was shown by Grechko, Belykh and Barabash (arXiv:2212.06672, 2022), via two auxiliary one-dimensional maps $f\pm\gamma$, to possess attractors in explicitly computable trapping domains for the quadratic nonlinearity $f(x)=\mu-x^2$ and the cubic $f(x)=x^3-\mu x$, and to possess a topological Smale horseshoe under explicit parameter inequalities in the quadratic case only. The constructive completion of that program is open: (i) for a general nonlinearity $f$, no criterion or procedure is known that decides whether trapping-interval endpoints $\alpha^\pm$ (and the coupled radius $\gamma$) satisfying the trapping condition exist, or computes them; (ii) the horseshoe-existence parameter inequalities are not available for cubic or higher-degree polynomial $f$ (the paper only remarks that similar arguments should apply).

Background

The Hénon map (M. Hénon, 1976) is the planar invertible map $(x,y)\mapsto(\mu-x^2+y,\,bx)$ introduced as a simple model with a strange attractor; the rigorous study of its attractor culminated in the Benedicks-Carleson analysis (1991). A natural multidimensional generalization, studied for instance by Gonchenko, Ovsyannikov, Simó and Turaev (2005) and by Gonchenko and Gonchenko (2016) in dimension three, replaces the planar map by an $(n+1)$-dimensional diffeomorphism whose linear part couples one 'nonlinear' direction with $n$ delay-like coordinates. Grechko, Belykh and Barabash (arXiv:2212.06672) study the concrete form

$\bar{x}=f(x)+\mathbb{1}y,\qquad \bar{y}=\mathbf{b}x+Ay,$

where $x\in\mathbb{R}$, $y\in\mathbb{R}^n$, $\mathbb{1}$ is the all-ones row (so $\mathbb{1}y=\sum_i y_i$), $\mathbf{b}=\mathrm{col}(b,0,\dots,0)$, and $A$ is the normalized lower-shift $n\times n$ matrix with entries $a_{ij}=a_j\,\delta_{i,j+1}$; the parameters satisfy $|b|<1$ and $|a_i|<a<1$. For $n=1$ and $f(x)=\mu-x^2$ this is the original Hénon map. A trapping region (invariant domain) is a set mapped into itself by the map; every orbit entering it remains inside, and the maximal invariant set it contains is an attractor. Using the Manhattan norm $\|y\|=\sum_i|y_i|$, in which the $y$-subsystem is contracting, the authors show (their Lemma 1) that if the slab $D_y=\{\|y\|<\gamma\}$ has radius $\gamma=\alpha|b|/(1-a)$ with $\alpha=\max\{|\alpha^-|,|\alpha^+|\}$, then the image of the box $D_\alpha=D_x\times D_y$, $D_x=(\alpha^-,\alpha^+)$, is confined by the two auxiliary one-dimensional maps $f^\pm(x)=f(x)\pm\gamma$. Their Theorem 1 states that if

$\alpha^-+\gamma\le f(x)\le \alpha^+-\gamma \quad\text{for all } x\in D_x,$

then the map has an attractor inside $D_\alpha$. Their Lemma 2 gives a partial sufficient condition of this kind: for a given $\gamma$, if an interval $I$ satisfies $\inf_{x\in I}f(x)-\gamma>\alpha^-$ and $\sup_{x\in I}f(x)+\gamma<\alpha^+$, then $(\alpha^-,\alpha^+)$ is a trapping domain with an attractor (equivalently, an invariant interval of the one-dimensional map $x\mapsto f(x)$ with a $\gamma$-margin supplies trapping endpoints). For the quadratic $f(x)=\mu-x^2$ (Theorem 2) and the odd cubic $f(x)=x^3-\mu x$ (Theorem 3) they solve these inequalities explicitly, obtaining closed-form domains and parameter regions. A Smale horseshoe is a hyperbolic invariant Cantor set produced by the transverse self-intersection of a stretched-and-folded strip, carrying a full shift as symbolic dynamics; a topological horseshoe is the corresponding crossing/covering structure that yields a semiconjugacy to a shift (in the sense of Kennedy and Yorke's topological horseshoes). Their Theorem 4 proves that in the parameter region $\sqrt{2(\mu+\gamma)}<\mu-\gamma$ the quadratic multidimensional map has a topological Smale horseshoe, the images of the lines $\{y=\mathrm{const}\}$ forming a solid folded arc crossing $D_x$ twice. Related earlier work (Li and Malkin 2006; Gonchenko, Li and Malkin 2008) established horseshoes for generalized Hénon maps only in perturbative regimes near a one-dimensional singular limit; the 2022 paper's point is to avoid smallness assumptions. Finally, for small coupling $|b|$ their Theorem 5 shows each structurally stable periodic orbit of the one-dimensional map $x\mapsto f(x)$ persists in the full map, with multipliers governed by an explicit characteristic polynomial (their equation (42)); after the cubic Theorem 3 they also identify one saddle-node bifurcation curve of the auxiliary maps (their equation (34)) and note it lies outside the invariant-domain region. What the paper does not do, and what remains open, is: solve the trapping-endpoint problem for a general $f$ (their Remark 1 states it explicitly) and extend the horseshoe inequalities to cubic and polynomial $f$ (their Remark 2 only asserts that similar arguments should apply).

Problem Statement

For the multidimensional Hénon map $\bar{x}=f(x)+\mathbb{1}y$, $\bar{y}=\mathbf{b}x+Ay$ with $x\in\mathbb{R}$, $y\in\mathbb{R}^n$, $\mathbb{1}$ the all-ones row, $\mathbf{b}=\mathrm{col}(b,0,\dots,0)$, $A$ the normalized lower-shift matrix ($a_{ij}=a_j\delta_{i,j+1}$), and parameters $|b|<1$, $|a_i|<a<1$, the open problem posed explicitly in Remarks 1 and 2 of arXiv:2212.06672 is to complete the constructive existence theory beyond the quadratic and attractor-only cubic cases. It has two linked components sharing one objective - making attractor and horseshoe existence for this map family constructive and general in the nonlinearity:

1. Trapping endpoints for general $f$. Given a scalar nonlinearity $f$ (continuous smooth or piecewise-smooth, in particular any polynomial, beyond the solved cases $f(x)=\mu-x^2$ and $f(x)=x^3-\mu x$), decide constructively whether distinct interval endpoints $\alpha^-<\alpha^+$ and a positive radius $\gamma$ exist such that the trapping condition $\alpha^-+\gamma\le f(x)\le\alpha^+-\gamma$ holds for all $x\in(\alpha^-,\alpha^+)$ and the box $D_\alpha=D_x\times D_y$ is a nonempty trapping domain, with $\gamma$ coupled to $\alpha=\max\{|\alpha^-|,|\alpha^+|\}$ through $\gamma=\alpha|b|/(1-a)$ (Lemma 1 of the source); when they exist, compute them; the target is a genuinely general decision/construction procedure.

2. Horseshoe conditions for polynomial $f$. Prove the existence of topological Smale horseshoes for the map with a general polynomial $f$ of arbitrary degree and real coefficients, the cubic $f(x)=x^3-\mu x$ being the first explicit case (which the general-polynomial result subsumes), under explicit parameter inequalities in the coefficients of $f$ and in the map parameters, analogous to the quadratic region $\sqrt{2(\mu+\gamma)}<\mu-\gamma$ (Theorem 4 of the source), including the verification that the image-arc crossing structure genuinely yields a (semi)conjugacy to a shift. This replaces the source's Remark 2, which asserts without proof or inequalities that similar arguments apply.

The target is, for the multidimensional Hénon map, what the source established only for $f(x)=\mu-x^2$ (and for the cubic attractor): constructive localization of attractors for general nonlinearity and explicit chaos-certifying parameter inequalities for polynomial nonlinearity.

References

1. D. A. Grechko, V. N. Belykh, N. V. Barabash, Existence of an attractor and Horseshoe in multidimensional Hénon map, arXiv:2212.06672 (2022), https://arxiv.org/abs/2212.06672
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