Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy

TL;DR

The paper addresses chaos generation in reversible dynamical systems by studying symmetric homoclinic tangles arising from a normally hyperbolic one-parameter family of symmetric periodic orbits with transversely intersecting stable and unstable manifolds. The authors develop a framework that reduces the flow near the tangle to a skew-product dynamics over a horseshoe, via invariant center laminations and a carefully constructed return map. They introduce scattering maps and an iterated-function-system analysis to prove positive topological entropy without requiring finite-type symbolic dynamics, and they establish the existence and regularity of a center lamination to support the reduced dynamics. The results apply to reversible systems beyond Hamiltonian settings and have implications for pattern formation in PDEs and non-holonomic mechanics, where symmetric tangles yield exponentially many patterns in finite windows and robust chaotic behavior.

Abstract

We consider reversible vector fields in $\mathbb{R}^{2n}$ such that the set of fixed points of the involutory reversing symmetry is $n$-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection.

Figures (6)

• Figure 1: Sketch of a homoclinic tangle, here depicted inside a three-dimensional cross-section $S$. The family $\{\gamma_a\}$ of symmetric periodic orbits, with $a$ from some interval $(a_-,a_+)$, intersects $\mathrm{Fix}\,R$ inside $S$, drawn as a solid curve. Each symmetric periodic orbit $\gamma_a$ has stable and unstable manifolds $V^s( \gamma_a)$, $V^u(\gamma_a)$. The intersections of these stable and unstable manifolds for the family $\{\gamma_a\}$ with $S$ form two-dimensional sheets: a stable sheet $V^s(\{\gamma_+a\}) \cap S$ bounded by $V^s( \gamma_{a_-})\cap S$ and $V^s( \gamma_{a_+})\cap S$, an unstable sheet $V^u(\{\gamma_+a\}) \cap S$ bounded by $V^u( \gamma_{a_-})\cap S$ and $V^u( \gamma_{a_+})\cap S$. The sheets $V^s( \{\gamma_a\})$, $V^u(\{\gamma_a\})$ intersect in $\mathrm{Fix}\,R$ inside $S$, not only in the family of periodic orbits, but also in a family of symmetric homoclinic points $\{\eta_a\} \cap S$. There are further intersections of the stable and unstable sheets outside $\mathrm{Fix}\,R$, a few are represented by dashed curves.
• Figure 2: Sketch of the lamination $\mathcal{F}^c$ near the leaves $\{p_a\}$ and $\{q_a\}$, that is invariant under an iterate $\Psi$ of $\Pi$.
• Figure 3: Sketch of nonsymmetric homoclinic leaves $\{r_a\}$ and $\{Rr_a\}$ and relevant maps between them.
• Figure 4: The scattering maps $\psi,\psi^{-1}$ define an iterated function system on an interval.
• Figure 5: The scattering maps $\psi,\psi^{-1}$ define an iterated function system on an interval, here in an example where it is necessary to restrict to a subinterval on which $\psi (u) \ne u$.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Proposition 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Lemma 2.6