Homoclinic tangencies of every order in the Hénon family

For the classical two-parameter Hénon family $f_{a,b}(x,y)=(1+y-ax^2,\;bx)$ of planar diffeomorphisms, Kiriki and Soma (2013), after proving that the family contains a generically unfolding cubic (order-2) homoclinic tangency at parameters arbitrarily close to $(-2,0)$ in the reparametrized coordinates $\varphi_{a,b}$ introduced below (the point $(2,0)$ in the classical $(a,b)$ coordinates), explicitly posed the question: does the original Hénon family have a homoclinic tangency of every finite order $n$? Order 1 (quadratic) and order 2 (cubic) are settled, but for every order $n\ge 3$ (quartic and higher) no existence result, impossibility result, or restatement is known in the later literature; the question remains open.

Background

The Hénon family, introduced by Hénon (1976) as a basic model of dissipative planar chaos, consists of the polynomial diffeomorphisms $f_{a,b}(x,y)=(1+y-ax^2,\;bx)$ with $b\neq 0$; Kiriki and Soma use the equivalent reparametrized form $\varphi_{a,b}(x,y)=(y,\;a-bx+y^2)$, obtained from the classical formula by $(a,b)\mapsto(-a,-b)$ and the linear coordinate change $(x,y)\mapsto(-ab^{-1}y,\;-ax)$. For $b\neq 0$ each $\varphi_{a,b}$ is a diffeomorphism of the plane, and its fixed points are $p_{a,b}^{\pm}=(y_{a,b}^{\pm},y_{a,b}^{\pm})$ with $y_{a,b}^{\pm}=\frac{1}{2}\bigl(1+b\pm\sqrt{(1+b)^2-4a}\bigr)$, so there are two of them whenever $(1+b)^2-4a>0$ (this discriminant is written in the $\varphi$-coordinates; the corresponding condition for the classical $f_{a,b}$ is $(1-b)^2+4a>0$). For $(a,b)$ sufficiently close to $(-2,0)$ with $b\neq 0$, both fixed points $p_{a,b}^{\pm}$ are saddles: the derivative at the fixed point has one eigenvalue $\lambda$ with $|\lambda|<1$ and one eigenvalue $\gamma$ with $|\gamma|>1$, and there $|\lambda\gamma|<1$, so both are dissipative saddles. A saddle then has a stable manifold $W^s(p)$ (the set of points whose forward orbits converge to $p$) and an unstable manifold $W^u(p)$ (points whose backward orbits converge to $p$), both injectively immersed curves. A homoclinic point is a point of $W^s(p)\cap W^u(p)$ other than $p$ itself; it is a homoclinic tangency when the two curves touch non-transversally there.

The order of a tangency measures the degree of contact: in local $C^{n+1}$ coordinates near the tangency in which the two manifold arcs are the graphs $y=0$ and $y=\alpha(x)$ with $\alpha(0)=0$, the tangency has order $n$ when $\alpha'(0)=\cdots=\alpha^{(n)}(0)=0$ and $\alpha^{(n+1)}(0)\neq 0$. Order 1 is called quadratic, order 2 cubic, order 3 quartic. A tangency in a family unfolds generically when the parameter moves the manifolds apart at nonzero speed: for a quadratic tangency in a one-parameter family this is $\partial_\mu\alpha(\mu_0,0)\neq 0$, and for a cubic tangency in a two-parameter family the analogous condition is the nonvanishing of a mixed parameter-derivative determinant. A saddle is dissipative when $|\lambda\gamma|<1$, so first returns near it contract area.

Homoclinic tangencies are the organizing centers of non-hyperbolic dynamics. Newhouse domains are open regions of the space of $C^r$ surface diffeomorphisms in which maps having homoclinic tangencies are dense. Gonchenko, Shilnikov and Turaev, in a long program on systems with structurally unstable Poincaré homoclinic orbits (see their Chaos 1996 survey and their Newhouse-domain papers), proved a striking hierarchy property in this setting: within Newhouse domains, diffeomorphisms having homoclinic tangencies of any prescribed finite order are dense (Gonchenko–Turaev–Shil'nikov 2001; Gonchenko–Turaev–Shilnikov 2007, the last also covering area-preserving maps). These, however, are genericity statements in the infinite-dimensional space of all surface diffeomorphisms, where one may freely perturb; they say nothing about any fixed finite-parameter family.

Kiriki and Soma (2008) showed what a generically unfolding cubic homoclinic tangency of a dissipative saddle implies for a two-parameter family: by renormalization (rescaling first-return maps near the tangency toward a one-dimensional limiting map), one obtains cubic dynamics — persistent antimonotonic tangencies (the concurrent creation and annihilation of periodic orbits along one-parameter paths, at cubic rather than quadratic degeneracies) and cubic polynomial-like strange attractors (attracting invariant sets equal to the closure of a saddle's unstable manifold on which some iterate is conjugate, up to scale, to a one-dimensional cubic polynomial map) supporting SRB measures, i.e. invariant measures with absolutely continuous conditional measures on unstable manifolds. Kiriki and Soma (2013) then instantiated the theory inside the Hénon family: using the Gonchenko–Shilnikov–Turaev mechanism, which creates a cubic homoclinic tangency from a cyclic pair of quadratic heteroclinic tangencies between two saddle fixed points, they proved (Theorem A) that there exists a parameter pair $(a_*,b_*)$ with $b_*>0$ arbitrarily close to $(-2,0)$ such that $\varphi_{a_*,b_*}$ has a cubic homoclinic tangency of its dissipative saddle fixed point $p^+_{a_*,b_*}$ that unfolds generically with respect to the two-parameter family, and they deduced (Corollary B) that the Hénon family exhibits cubic dynamics.

For order 1 the situation is classical: quadratic homoclinic tangencies of the Hénon map at explicit parameter values, unfolding generically in one-parameter subfamilies, have been certified by the computer-assisted proof of Wilczak and Zgliczyński (2009), and quadratic tangencies are ubiquitous in the family's non-hyperbolic parameter regions. The Gonchenko–Shilnikov–Turaev density theorems leave open, however, whether the specific two-parameter Hénon family itself realizes tangencies of arbitrarily high order, since a two-parameter slice cannot inherit a density statement that requires arbitrarily many independent perturbation directions. This is precisely the gap Kiriki and Soma pointed at immediately after Theorem A.

Problem Statement

Does the classical Hénon family $f_{a,b}(x,y)=(1+y-ax^2,\;bx)$, $b\neq 0$ (equivalently, in the reparametrized form $\varphi_{a,b}(x,y)=(y,\;a-bx+y^2)$ used in the source work), contain a homoclinic tangency of every finite order? That is, for each integer $n\ge 1$, is there a parameter value $(a,b)$ such that the corresponding map has a saddle fixed point $p$ whose stable and unstable manifolds have a homoclinic tangency of order $n$ — in local $C^{n+1}$ coordinates near the tangency with the manifold arcs given as $y=0$ and $y=\alpha(x)$, the conditions $\alpha(0)=\alpha'(0)=\cdots=\alpha^{(n)}(0)=0$ and $\alpha^{(n+1)}(0)\neq 0$? Orders $n=1$ (quadratic) and $n=2$ (cubic, near $(-2,0)$ in the $\varphi$-coordinates) are known; the open core is every order $n\ge 3$ (quartic and higher), with no restriction on the region of the $(a,b)$-parameter plane. No generic-unfolding requirement is imposed for $n\ge 3$: the source defines generic unfolding only for a quadratic tangency in a one-parameter family and for a cubic tangency in a two-parameter family, so generic unfolding enters only as part of the known results at orders $n=1$ and $n=2$.

References

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