Dissecting a Resonance Wedge on Heteroclinic Bifurcations

Abstract

This article studies routes to chaos occurring within a resonance wedge for a 3-parametric family of differential equations acting on a 3-sphere. Our starting point is an autonomous vector field whose flow exhibits an attracting heteroclinic network made by two 1-dimensional connections and a 2-dimensional separatrix between two equilibria with different Morse indices. After changing the parameters, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We derive the first return map near the network and we reduce the analysis of the system to a 2-dimensional map on the cylinder. Complex dynamical features arise from a discrete-time Bogdanov–Takens singularity, which may be seen as the organizing center by which one can obtain infinitely many attracting tori, strange attractors, infinitely many sinks and non-trivial contracting wandering domains. These dynamical phenomena occur within a structure that we call resonance wedge. As an application, we may see the “classical” Arnold tongue as a projection of a resonance wedge. The results are general, extend to other contexts and lead to a fine-tuning of the theory.

Appendix A. Glossary

We record a miscellaneous collection of terms and terminology that are used throughout the text. For $\varepsilon >0$ small, consider the 3-parameter family of $C^3$–smooth autonomous differential equations

$$\begin{aligned} \dot{x}=f_{(A, \lambda , \omega )}(x)\qquad x\in {\mathbb {S}}^3\subset {\mathbb {R}}^4 \qquad A, \lambda \in [0, \varepsilon ], \qquad \omega \in {\mathbb {R}}^+. \end{aligned}$$

(A.1)

Since ${\mathbb {S}}^3$ is a compact set without boundary, the local solutions of (A.1) could be extended to ${\mathbb {R}}$. Denote by $\varphi _{(A, \lambda , \omega )}(t,x)$, $t \in {\mathbb {R}}$, the associated flow.

1.1 Symmetry

Given a compact Lie group $\mathcal {G}$ of endomorphisms of ${\mathbb {R}}^4$, we will consider 3-parameter families of vector fields $(f_{(A, \lambda , \omega )})$ under the equivariance assumption

$$\begin{aligned} f_{(A, \lambda , \omega )}(\gamma x)=\gamma f_{(A, \lambda , \omega )}(x) \end{aligned}$$

for all $x \in {\mathbb {S}}^3$, $\gamma \in \mathcal {G}$ and $(A, \lambda , \omega )\in [0, \varepsilon ]^2\times {\mathbb {R}}^+.$ For an isotropy subgroup $\widetilde{\mathcal {G}}< \mathcal {G}$, we write ${\text {Fix}}(\widetilde{\mathcal {G}})$ for the vector subspace of points that are fixed by the elements of $\widetilde{\mathcal {G}}$. Note that, for $\mathcal {G}-$equivariant differential equations, the subspace ${\text {Fix}}(\widetilde{\mathcal {G}})$ is flow-invariant.

1.2 Attracting Set

A subset $\Omega$ of ${\mathbb {S}}^3$ for which there exists a neighborhood $U \subset {\mathbb {S}}^3$ satisfying $\varphi _{(A, \lambda , \omega )}(t,U)\subset U$ for all $t\ge 0$ and

$$\begin{aligned} \displaystyle \bigcap _{t\,\in \,{\mathbb {R}}^+}\,\varphi _{(A, \lambda , \omega )}(t,U)=\Omega \end{aligned}$$

is called an attracting set by the flow of (A.1). This set is not necessarily connected. Its basin of attraction, denoted by $\mathbf{B} (\Omega )$, is the set of points in ${\mathbb {S}}^3$ whose orbits have $\omega -$limit in $\Omega$. We say that $\Omega$ is asymptotically stable (or $\Omega$ is a global attractor) if $\mathbf{B} (\Omega )= {\mathbb {S}}^3$. An attracting set is said to be quasi-stochastic if it encloses periodic solutions with different Morse indices (dimension of the unstable manifold), structurally unstable cycles, sinks and saddle-type invariant sets (cf. [24]).

1.3 Heteroclinic Structures

Suppose that $O_1$ and $O_2$ are two hyperbolic equilibria of (A.1) with different Morse indices (dimension of the unstable manifold). There is a heteroclinic cycle associated to $O_1$ and $O_2$ if

$$\begin{aligned} W^{u}(O_1)\cap W^{s}(O_2)\ne \emptyset \qquad \text {and} \qquad W^{u}(O_2)\cap W^{s}(O_1)\ne \emptyset . \end{aligned}$$

For $i\ne j \in \{1,2\}$, the non-empty intersection of $W^{u}(O_i)$ with $W^{s}(O_j)$ is called a heteroclinic connection between $O_i$ and $O_j$, and will be denoted by $[O_i \rightarrow O_j]$. Although heteroclinic cycles involving equilibria are not a generic property within differential equations, they may be structurally stable within families of vector fields which are equivariant under the action of a compact Lie group $\mathcal {G}\subset \mathbb {O}(4)$, due to the existence of flow-invariant subspaces [26].

A heteroclinic cycle between two hyperbolic saddle-foci of different Morse indices, where one of the connections is transverse while the other is structurally unstable, is called a Bykov cycle. We address the reader to [30] for an overview of heteroclinic bifurcations and substantial information on the dynamics near different types of structures.

1.4 Historic Behaviour

We say that the solution of (A.1), $\varphi _{(A, \lambda , \omega )}(t,x)$ with $x \in {\mathbb {S}}^3$, has historic behaviour if there is a continuous function ${H}:{\mathbb {S}}^3\rightarrow {\mathbb {R}}$ such that the time average $\displaystyle \frac{1}{T}\int _{0}^{T} {H} (\varphi _{(A, \lambda , \omega )}(t,x)) dt\, \,$ fails to converge.

1.5 Strange Attractor

A (Hénon-type) strange attractor of a two-dimensional dissipative diffeomorphism R defined in a Riemannian manifold $\mathcal {M}$, is a compact invariant set $\Lambda$ with the following properties:

• $\Lambda$ equals the closure of the unstable manifold of a hyperbolic periodic point;

• the basin of attraction of $\Lambda$ contains an open set;

• there is a dense orbit in $\Lambda$ with a positive Lyapounov exponent (exponential growth of the derivative along its orbit).

A vector field possesses a strange attractor if the first return map to a cross section does.

1.6 SRB Measure

Given an attracting set ${\Omega }$ for a continuous map $R: \mathcal {M} \rightarrow \mathcal {M}$ where $\mathcal {M}$ is a compact smooth manifold, consider the Birkhoff average with respect to the continuous function $T: \mathcal {M} \rightarrow {\mathbb {R}}$ on the R-orbit starting at $x\in \mathcal {M}$:

$$\begin{aligned} L(T, x)=\lim _{n\in {\mathbb {N}}} \quad \frac{1}{n} \sum _{i=0}^{n-1} T \circ R^i(x). \end{aligned}$$

(A.2)

Suppose that, for Lebesgue almost all points $x\in \mathbf{B} ({\Omega })$, the limit (A.2) exists and is independent on x. Then L is a continuous linear functional in the set of continuous maps from $\mathcal {M}$ to ${\mathbb {R}}$ (denoted by $C(\mathcal {M}, {\mathbb {R}})$). By the Riesz Representation Theorem, it defines a unique probability measure $\mu$ such that:

$$\begin{aligned} \lim _{n\in {\mathbb {N}}} \quad \frac{1}{n} \sum _{i=0}^{n-1} T \circ R^i(x) = \int _{\Omega } T \, d\mu \end{aligned}$$

(A.3)

for all $T\in C(\mathcal {M}, {\mathbb {R}})$ and for Lebesgue almost all points $x\in \mathbf{B} ({\Omega })$. If there exists an ergodic measure $\mu$ supported in ${\Omega }$ such that (A.3) is satisfied for all continuous maps $T\in C(\mathcal {M}, {\mathbb {R}})$ for Lebesgue almost all points $x\in \mathbf{B} ({\Omega })$, where $\mathbf{B} ({\Omega })$ has positive Lebesgue measure, then $\mu$ is called a SRB (Sinai-Ruelle-Bowen) measure and ${\Omega }$ is a SRB attractor. More details in [55].

1.7 Non-trivial Wandering Domains

A non-trivial wandering domain for a given map R on a Riemannian manifold $\mathcal {M}$ is a non-empty connected open set $D \subset \mathcal {M}$ which satisfies the following conditions:

• $R^i(D)\cap R^j(D)=\emptyset$ for every $i,j\ge 0$ ($i\ne j$)

• the union of the $\omega$-limit sets of points in D for R, denoted by $\Omega (D,R)$, is not equal to a single periodic orbit.

A wandering domain D is called contracting if the diameter of $R^n(D)$ converges to zero as $n \rightarrow +\infty$.

1.8 Rotational Horseshoe

Let $\mathcal {H}$ stand for the infinite annulus $\mathcal {H} = {\mathbb {S}}^1 \times {\mathbb {R}}$ (endowed with the usual inner product from ${\mathbb {R}}^2$). We denote by $Homeo^+(\mathcal {H} )$ the set of homeomorphisms of the annulus which preserve orientation. Given a homeomorphism $f :X \rightarrow X$ and a partition of $m\in {\mathbb {N}}\backslash \{1\}$ elements $R_0,..., R_{m-1}$ of $X\subset \mathcal {H}$, the itinerary function $\xi : X \rightarrow \{0, ..., m-1\}^{\mathbb {Z}}= \Sigma _m$ is defined by:

$$\begin{aligned} \xi (x)(j)=k\quad \Leftrightarrow \quad f^j(x)\in R_k, \quad \text {for every} \quad j\in {\mathbb {Z}}. \end{aligned}$$

Following [43], we say that a compact invariant set $\Lambda \subset \mathcal {H}$ of $f \in Homeo^+(\mathcal {H} )$ is a rotational horseshoe if it admits a finite partition $P =\{R_0, ..., R_{m-1} \}$ by sets $R_i$ with non empty interior in $\Lambda$ so that:

• the itinerary $\xi$ defines a semi-conjugacy between $f|_\Lambda$ and the full-shift $\sigma : \Sigma _m \rightarrow \Sigma _m$, that is $\xi \circ f = \sigma \circ \xi$ with $\xi$ continuous and onto;

• for any lift $F: {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2$ of f, there exist $k>0$ and m vectors $v_0, ...,v_{m-1} \in {\mathbb {Z}}\times \{0\}$ so that:

  $$\begin{aligned} \left\| (F^n(\hat{x})-\hat{x}) - \sum _{i=0}^n v_{\xi (x)(i)}\right\| <k \qquad \text {for every} \qquad \hat{x}\in \pi ^{-1}(\Lambda ), \quad n\in {\mathbb {N}}, \end{aligned}$$

  where $\Vert \star \Vert$ is the usual norm on ${\mathbb {R}}^2$, $\pi :{\mathbb {R}}^2\rightarrow \mathcal {H}$ denotes the usual projection map and $\hat{x} \in \pi ^{-1}(\Lambda )$ is the lift of x; more details in the proof of Lemma 3.1 of [43]. The existence of a rotational horseshoe for a map implies positive topological entropy at least $\log m$.