Transitions of bifurcation diagrams of a forced heteroclinic cycle

TL;DR

This work analyzes how a periodically forced, attracting robust heteroclinic cycle on $S^2$ reorganizes its dynamics when subjected to time-periodic forcing. By reducing the flow near the cycle to a two-dimensional circle-map framework and introducing a lag parameter $\tau$, the authors classify all persistent bifurcation diagrams on a cylinder, revealing a sharp golden-ratio threshold $\Phi=(1+\sqrt{5})/2$ that separates weak and strong attraction regimes. They prove the existence of a codimension-2 discrete-time Bogdanov–Takens bifurcation, a Hopf surface, and a network of saddle-node, Hopf, and homoclinic bifurcations, which collectively underpin frequency locking, invariant tori, and chaotic dynamics (suspended horseshoes) near heteroclinic tangencies. The results connect to higher-dimensional phenomena and offer analytic explanations for long-period, frequency-locked, and chaotic regimes observed in forced heteroclinic networks, with implications for models of winnerless competition and related systems.

Abstract

A family of periodic perturbations of an attracting robust heteroclinic cycle defined on the two-sphere is studied by reducing the analysis to that of a one-parameter family of maps on a circle. The set of zeros of the family forms a bifurcation diagram on the cylinder. The different bifurcation diagrams and the transitions between them are obtained as the strength of attraction of the cycle and the amplitude of the periodic perturbation vary. We determine a threshold in the cycle's attraction strength above which frequency locked periodic solutions with arbitrarily long periods bifurcate from the cycle as the period of the perturbation decreases. Below this threshold further transitions are found giving rise to a frequency locked invariant torus and to a frequency locked suspended horseshoe, arising from heteroclinic tangencies in the family of maps.

Figures (10)

• Figure 1: Graph of $\tau\to{\mathcal{F}}_\delta(\tau)$ for different values of $\delta$. On the left the solid line is the graph for $1<\delta<\Phi$, the dashed line is the graph of ${\mathcal{F}}_\Phi(\tau)$. On the right the solid line is the graph for $\delta>\Phi$, the dashed line is the graph of ${\mathcal{F}}_\Phi(\tau)$.
• Figure 2: Graph of $s\mapsto \gamma(1+k\sin s)$ for $k<1$ (left) and $k>1$ (right).
• Figure 3: Illustration of Theorem \ref{['th:k>1']}. Top: persistent bifurcation diagrams on the cylinder with $k>1$, labelling of regions as in Theorem \ref{['th:k>1']}. Bottom: transition diagram for the map \ref{['eq:g0']} with $k>1$ and bifurcation diagrams for parameters on the curves separating the regions. The horizontal boundaries in the bifurcation diagrams are identified, we indicate this by the line /.
• Figure 4: Fold points for the map \ref{['eq:g0']} with $k>1$. Solid lines are supercritical folds, dashed lines are subcritical.
• Figure 5: Illustration of Theorem \ref{['th:k<1']}. Top: transition diagram for the map $g$ defined in \ref{['eq:g0']} with $0<k<1$. In the region W there are no zeros of the map $g$. Center: persistent bifurcation diagrams. Bottom: bifurcation diagrams for parameters in the curves separating the regions. For $\delta=\Phi$ a branch of the diagram is absorbed by the boundary $\tau=0$. The horizontal boundaries in the bifurcation diagrams are identified, we indicate this by the line /.

Theorems & Definitions (30)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4