Three-dimensional horseshoes near an unfolding of a Hopf-Hopf singularity

TL;DR

This work analyzes unfoldings of a Hopf–Hopf singularity in a four-dimensional vector field and identifies a heteroclinic network consisting of a bifocus and two hyperbolic periodic orbits. By constructing detailed local and global transition maps and verifying Conley–Moser-type conditions, it proves the existence of infinitely many linked tridimensional hyperbolic horseshoes that accumulate on the heteroclinic network for small $\gamma>0$, and shows abrupt heteroclinic switching and symbolic dynamics on a Cantor invariant set. The results are specialized to a Gaspard-type unfolding, where the hyperbolic horseshoes persist under a concrete analytic perturbation, with order-5 truncation being minimal to guarantee the organizing center’s chaotic behavior. The findings provide a rigorous mechanism for chaotic dynamics near Hopf–Hopf unfoldings and set the stage for exploring persistent strange attractors and tangencies in higher-dimensional heteroclinic networks.

Abstract

Motivated by a certain type of unfolding of a Hopf-Hopf singularity, we consider a one-parameter family $(f_γ)_{γ\geq0}$ of $C^3$--vector fields in $\mathbb{R}^4$ whose flows exhibit a heteroclinic cycle associated to two periodic solutions and a bifocus, all of them hyperbolic. It is formally proved that combining rotation with a generic condition concerning the transverse intersection between the three-dimensional invariant manifolds of the periodic solutions, all sets are highly distorted by the first return map and hyperbolic three-dimensional horseshoes emerge, accumulating on the network. Infinitely many linked horseshoes prompt the coexistence of infinitely many saddle-type invariant sets for all values of $γ\gtrsim 0$ belonging to the heteroclinic class of the two hyperbolic periodic solutions. We apply the results to a particular unfolding of the Hopf-Hopf singularity, the so called \emph{Gaspard-type unfolding}.

Figures (13)

• Figure 1: (a) Heteroclinic cycle associated to $\mathcal{O}\to (0,0)$$\mathbf{E}_1\to (-\mu_1,0)$ and $\mathbf{E}_2\to (0, \mu_2)$. The cycle is attracting by inside. (b) Bifurcation diagram of \ref{['HH_amplitude']} in the parameters $(\mu_1,\mu_2)\in {\mathbf R}^2$. The curve Het is the graph of the expression \ref{['curve1']}.
• Figure 2: Coordinates and cross-sections near the bifocus $\bold{O}$. (a): $\Sigma_0^{in}$ and (b): $\Sigma_0^{out}$. The superscripts "in" and "out" have been omitted in order to lighten the figures.
• Figure 3: Coordinates and cross-sections near the periodic solution ${\mathbf C}_1$. (a): $\Sigma_1^{in}$ and (b): $\Sigma_1^{out}$. The superscripts "in" and "out" have been omitted in order to lighten the figures.
• Figure 4: Illustration of the intersections of $W^u(\mathcal{C}_2)$ and $W^s(\mathcal{C}_1)$ with $\Sigma_2^{out}$. The corresponding two-dimensional tori unfold from a coincidence, intersecting along two circles, $\ell_1^{out}$ and $\ell_2^{out}$. Extending these circles by the flow one generates the tubular sets $\mathcal{T}_1$ and $\mathcal{T}_2$. For greater clarity, a rectangle $P$, transverse to the tori, is included; the black circles in $P$ represent the intersections of the tori with $P$.
• Figure 5: (a): Representation of $C_1^{in}$ and $C_2^{in}$ as subsets of $\Sigma_1^{in}$. (b): Representation of $C_1^{out}$ and $C_2^{out}$ as subsets of $\Sigma_2^{out}$.

Theorems & Definitions (43)

• Remark 1
• Proposition 1
• Theorem 1
• Definition 1
• Definition 2
• Definition 3
• Corollary 2
• Definition 4
• Corollary 3
• Theorem 4: Shilnikov et al Shilnikov et al, adapted