Unfolding a Bykov Attractor: From an Attracting Torus to Strange Attractors

Abstract

In this paper we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two hyperbolic saddles-foci with different Morse indices. After slightly increasing both parameters, while keeping the one-dimensional connections unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the equilibria do not intersect. Under some conditions on the parameters and on the eigenvalues of the linearisation of the vector field at the saddle-foci, we prove the existence of many complicated dynamical objects, ranging from an attracting quasi-periodic torus to Hénon-like strange attractors, as a consequence of the Torus-Breakdown Theory. The mechanism for the creation of horseshoes and strange attractors is also discussed. Theoretical results are applied to show the occurrence of strange attractors in some analytic unfoldings of a Hopf-zero singularity.

Glossary

For $\varepsilon >0$ small, consider the two-parameter family of $C^3$-smooth autonomous differential equations

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

(A.1)

Denote by $\varphi _{(A, \lambda )}(t,x)$, $t \in {\mathbb {R}}$, the associated flow.

1.1 Symmetry

Given a group $\mathcal {G}$ of endomorphisms of ${\mathbb {S}}^3$, we will consider two-parameter families of vector fields $(f_{(A, \lambda )})$ under the equivariance assumption $f_{(A, \lambda )}(\gamma x)=\gamma f_{(A, \lambda )}(x)$ for all $x \in {\mathbb {S}}^3$, $\gamma \in \mathcal {G}$ and $(A, \lambda )\in [0, \varepsilon ]^2.$ For an isotropy subgroup $\widetilde{\mathcal {G}}< \mathcal {G}$, we will write ${\text {Fix}}(\widetilde{\mathcal {G}})$ for the vector subspace of points that are fixed by the elements of $\widetilde{\mathcal {G}}$. Observe 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 a topological space $\mathcal {M}$ for which there exists a neighborhood $U \subset \mathcal {M}$ satisfying $\varphi (t,U)\subset U$ for all $t\ge 0$ and $\bigcap _{t\,\in \,{\mathbb {R}}^+}\,\varphi (t,U)=\Omega$ is called an attracting set by the flow $\varphi$, not necessarily connected. Its basin of attraction, denoted by $\mathbf{B} (\Omega )$ is the set of points in $\mathcal {M}$ whose orbits have $\omega -$limit in $\Omega$. We say that $\Omega$ is asymptotically stable (or that $\Omega$ is a global attractor) if $\mathbf{B} (\Omega )=\mathcal {M}$. An attracting set is said to be quasi-stochastic if it encloses periodic solutions with different Morse indices, structurally unstable cycles, sinks and saddle-type invariant sets.

1.3 Heteroclinic Structures

Suppose that $O_1$ and $O_2$ are two hyperbolic saddle-foci of $f_{(A, \lambda )}$ with different Morse indices (dimension of the unstable manifold). There is a heteroclinic cycle associated to $O_1$ and $O_2$ if $W^{u}(O_1)\cap W^{s}(O_2)\ne \emptyset$ and $W^{u}(O_2)\cap W^{s}(O_1)\ne \emptyset .$ For $i, j \in \{1,2\}$, 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 feature within differential equations, they may be structurally stable within families of systems which are equivariant under the action of a compact Lie group $\mathcal {G}\subset \mathbb {O}(n)$, due to the existence of flow-invariant subspaces [25].

1.4 Bykov Cycle

A heteroclinic cycle between two hyperbolic saddle-foci of different Morse indices, where one of the connections is transverse (and so stable under small perturbations) while the other is structurally unstable, is called a Bykov cycle. A Bykov network is a connected union of heteroclinic cycles, not necessarily in finite number. We refer to [29] for an overview of heteroclinic bifurcations and substantial information on the dynamics near different kinds of heteroclinic cycles and networks.

1.5 Suspended Horseshoe

Given $(A, \lambda ) \in [0, \varepsilon ]^2$, suppose that there is a cross-section $\mathcal {S}_\lambda$ to the flow $\varphi _{(A, \lambda )}$ such that $\mathcal {S}_{(A, \lambda )}$ contains a compact set $\mathcal {K}_{(A, \lambda )}$ invariant by the first return map $\mathcal {F}_{(A, \lambda )}$ to $\mathcal {S}_{(A, \lambda )}$. Assume also that $\mathcal {F}_{(A, \lambda )}$ restricted to $\mathcal {K}_{(A, \lambda )}$ is conjugate to a full shift on a finite alphabet. Then the suspended horseshoe associated to $\mathcal {K}_{(A, \lambda )}$ is the flow-invariant set $\widetilde{\mathcal {K}_{(A, \lambda )}}=\{\varphi _\lambda (t,x)\,:\,t\in {\mathbb {R}},\,x\in \mathcal {K}_{(A, \lambda )}\}.$

1.6 SRB Measure

Given an attracting set ${\Omega }$ for a continuous map $R: \mathcal {M} \rightarrow \mathcal {M}$ of a compact manifold $\mathcal {M}$, 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 measure and ${\Omega }$ is a SRB attractor.