The Role of the Saddle-Foci on the Structure of a Bykov Attracting Set

Abstract

We consider a one-parameter family $(f_\lambda )_{\lambda \, \geqslant \, 0}$ of symmetric vector fields on the three-dimensional sphere whose flows exhibit a heteroclinic network between two saddle-foci inside a global attracting set. More precisely, when $\lambda = 0$, there is an attracting heteroclinic cycle between the two equilibria which is made of two 1-dimensional connections together with a 2-dimensional sphere which is both the stable manifold of one saddle-focus and the unstable manifold of the other. After slightly increasing the parameter while keeping the 1-dimensional connections unaltered, the two-dimensional invariant manifolds of the equilibria become transversal, and thereby create homoclinic and heteroclinic tangles. It is known that these newborn structures are the source of a countable union of topological horseshoes, which prompt the coexistence of infinitely many sinks and saddle-type invariant sets for many values of $\lambda$. We show that, for every small enough positive parameter $\lambda$, the stable and unstable manifolds of the saddle-foci and those infinitely many horseshoes are contained in the global attracting set of $f_\lambda$; moreover, the horseshoes belong to the heteroclinic class of the equilibria. In addition, we show that the set of chain-accessible points from either of the saddle-foci is chain-stable and contains the closure of the invariant manifolds of the two equilibria.

Appendix A. Glossary

1.1 A.1. Hyperbolicity

Let M be a smooth compact Riemannian manifold, and consider a diffeomorphism $h: M\rightarrow M$ and a compact h-invariant set ${\mathcal {K}} \subset {\mathbb {S}}^3$. We say that ${\mathcal {K}}$ is uniformly hyperbolic if there are constants $C>0$ and $0< \mu < 1$ such that, for every $x \in {\mathcal {K}}$, there is a splitting of the tangent space $T_x M=E^s_x\oplus E^u_x$ satisfying, for every $n\in {\mathbb {N}}$,

$$\begin{aligned} \Vert Dh^n(v)\Vert \leqslant C \mu ^n \Vert v\Vert \quad \forall \, v\in E^s_x \qquad \text {and} \qquad \Vert Dh^{-n}(v)\Vert \leqslant C \mu ^n \Vert v\Vert \quad \forall \, v\in E^u_x. \end{aligned}$$

1.2 A.2. Symmetry

Given a group ${\mathcal {G}}$ of endomorphisms of ${\mathbb {S}}^3$, we say that a one-parameter family of vector fields $(f_\lambda )_{\lambda \,\in \,{\mathbb {R}}}$ is symmetric it it satisfies the equivariance assumption

$$\begin{aligned} f_\lambda (\gamma x)=\gamma f_\lambda (x) \quad \quad \forall \,x \in {\mathbb {S}}^3, \quad \forall \,\gamma \in {\mathcal {G}}\, \text { and }\, \forall \,\lambda \in {\mathbb {R}}. \end{aligned}$$

1.3 A.3. Attracting Set

A subset A of a topological space ${\mathcal {M}}$ is said to be attracting by a flow $\varphi$ if there exists an open set $U \subset {\mathcal {M}}$ satisfying

$$\begin{aligned} \varphi (t,U)\subset U \quad \quad \forall \,t \geqslant 0 \, \text { and }\, \bigcap _{t\,\in \,{\mathbb {R}}^+}\,\varphi (t,U)=A. \end{aligned}$$

Its basin of attraction, denoted by ${\mathcal {B}}(A)$, is the set of points in ${\mathcal {M}}$ whose orbits have $\omega$-limit in A. We say that A is asymptotically stable (or, equivalently, that A is a global attracting set) if ${\mathcal {B}}(A) = {\mathcal {M}}$. An attracting set is said to be quasi-stochastic if it encloses periodic trajectories with different Morse indices (the Morse index of a hyperbolic equilibrium is the dimension of its unstable manifold), structurally unstable cycles, sinks and saddle-type invariant sets.

1.4 A.4. Heteroclinic Phenomena

Suppose that $\sigma _1$ and $\sigma _2$ are two hyperbolic saddle-foci of a vector field f with different Morse indices. We say that there is a heteroclinic cycle associated to $\sigma _1$ and $\sigma _2$ if $W^u(\sigma _1)\cap W^s(\sigma _2)\ne \emptyset$ and $W^u(\sigma _2)\cap W^s(\sigma _1)\ne \emptyset .$ For $i, j \in \{1,2\}$, the non-empty intersection of $W^u(\sigma _i)$ with $W^s(\sigma _j)$ is called a heteroclinic connection between $\sigma _i$ and $\sigma _j$, and will be denoted by $[\sigma _i \rightarrow \sigma _j]$. Although heteroclinic cycles involving equilibria are not a generic feature within differential equations, they may be structurally stable within families of vector fields which are equivariant under the action of a compact Lie group ${\mathcal {G}}\subset \mathbf{O }(n)$, due to the existence of flow-invariant subspaces [9]. A heteroclinic network is a connected finite union of heteroclinic cycles.

1.5 A.5. Bykov Cycle

A heteroclinic cycle between two hyperbolic saddle-foci of a vector field f with 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.

1.6 A.6. Tubular Neighborhood of a Bykov Cycle

Given a Bykov cycle associated to $\sigma _1$ and $\sigma _2$, let $V_1, V_2$ be two small and disjoint cylindrical neighborhoods of $\sigma _1$ and $\sigma _2$, respectively. Consider two local sections transverse to the cycle at two points $p_1$ and $p_2$ in the connections $[\sigma _1\rightarrow \sigma _2]$ and $[\sigma _2\rightarrow \sigma _1]$, respectively, with $p_1, p_2 \not \in V_1\cup V_2$. Saturating the cross sections by the flow, we obtain two flow-invariant tubes joining $V_1$ and $V_2$ which contain the connections in their interior. The union of those tubes with $V_1$ and $V_2$ is called a tubular neighborhood of the Bykov cycle.

1.7 A.7. Chirality

There are two different possibilities for the geometry of a flow around a Bykov cycle $\Gamma$, depending on the direction the trajectories turn around the one-dimensional heteroclinic connection between $\sigma _1$ and $\sigma _2$. Let $V_1$ and $V_2$ be small disjoint neighborhoods of $\sigma _1$ and $\sigma _2$ with disjoint boundaries $\partial V_1$ and $\partial V_2$, respectively. Trajectories starting at ${\text {In}}(\sigma _1) {\setminus } W^s(\sigma _1)$ enter $V_1$ in positive time, then follow the connection from $\sigma _1$ to $\sigma _2$, enter $V_2$, finally leaving $V_2$ at $\partial V_2$. Let $\gamma$ be a piece of trajectory like the one just described from ${\text {In}}(\sigma _1) {\setminus } W^s(\sigma _1)$ to $\partial V_2$. Now join its starting point to its end point by a line segment (see Figure 1 of [14]), forming a closed curve that we call the loop of$\gamma$. The Bykov cycle $\Gamma$ and the loop of $\gamma$ are disjoint closed sets. We say that the two saddle-foci $\sigma _1$ and $\sigma _2$ in $\Gamma$ have the same chirality if the loop of every such a trajectory is linked to $\Gamma$, in the sense that these two sets cannot be disconnected by an isotopy in ${\mathbb {R}}^{4}$.

1.8 A.8. Pulse

Let $V_1, V_2$ be two small and disjoint neighborhoods of $\sigma _1$ and $\sigma _2$, respectively, and take $n \in {\mathbb {N}}$. A one-dimensional heteroclinic connection from $\sigma _2$ to $\sigma _1$ which, after leaving $V_2$, enters and leaves both $V_1$ and $V_2$ precisely n times is called an n-pulse heteroclinic connection with respect to $V_1$ and $V_2$, or simply an n-pulse. A 0-pulse is a one-dimensional heteroclinic connection from $\sigma _2$ to $\sigma _1$ which, after leaving $V_2$, enters $V_1$ and afterwards stays in this neighborhood.

1.9 A.9. Saturated Set and the Operator Tilde

Let ${\mathcal {S}}$ be a cross-section to a flow $\varphi$ and assume that ${\mathcal {S}}$ contains a compact set ${\mathcal {K}}$ invariant by the first return map ${\mathcal {R}}$ to ${\mathcal {S}}$. Then the saturation of${\mathcal {K}}$, we denote by $\widetilde{{\mathcal {K}}}$, is the flow-invariant set formed by the $\varphi$-trajectories of points of ${\mathcal {K}}$, that is, $\Big \{\varphi (t,x)\,:\,t\,\in \,{\mathbb {R}},\,\,x\,\in \, {\mathcal {K}}\Big \}.$

1.10 A.10. Chain-Accessible Point

Given $\varepsilon >0$ and $\tau > 0$, an $(\varepsilon ,\tau )$-trajectory of a flow $\varphi$ is a finite set $\{P_1,\ldots ,P_k\}$ such that, for all $i\in \{1, \ldots , k-1 \}$, the point $P_{i+1}$ is at a distance strictly smaller than $\varepsilon$ from $\varphi (t, P_i)$ for some $t > \tau$. A point Q is said to be $(\varepsilon ,\tau )$-chain-accessible from a point P if there exists an $(\varepsilon ,\tau )$-trajectory $\{P_1, P_2,\ldots ,P_k\}$connectingP with Q; in particular, P and Q are at a distance strictly smaller than $\varepsilon$ from $P_1$ and $P_k$, respectively. A point Q is said to be chain-accessible from a point P if, for any $\varepsilon > 0$, there exists $\tau > 0$ such that Q is $(\varepsilon ,\tau )$-chain-accessible from P.

1.11 A.11. Chain-Recurrent Class

A point P is called chain-recurrent for a flow $\varphi$ if it is chain-accessible from $\varphi (t,P)$ for any $t \in {\mathbb {R}}$. More generally, a set ${\mathcal {C}}$ is chain-accessible from a point P if it contains a point chain-accessible from P. A compact invariant set ${\mathcal {C}}$ is a chain-recurrent class if, given any two points in ${\mathcal {C}}$ and $\varepsilon >0$, we may find an $(\varepsilon ,\tau )$-trajectory connecting them entirely contained in ${\mathcal {C}}$. We observe that, by definition, every point of a chain-recurrent class is chain-recurrent. A set ${\mathcal {C}}$ is called chain-stable if, for any neighborhood ${\mathcal {V}}$ of ${\mathcal {C}}$, there exists an $\varepsilon >0$ such that every $(\varepsilon , \tau )$-trajectory connecting any two points of ${\mathcal {C}}$ is contained in ${\mathcal {V}}$.