Bifurcations in the Kuramoto model with external forcing and higher-order interactions

TL;DR

This work investigates a Kuramoto ensemble subjected to external forcing and higher-order (simplicial) interactions, revealing an exceptionally rich bifurcation structure. Using the Ott–Antonsen ansatz and a Lorentzian frequency distribution, the authors reduce the dynamics to two order-parameter equations and map saddle-node, Hopf, and homoclinic bifurcations across parameter spaces. They show that higher-order couplings can duplicate bifurcation manifolds and induce multi-stability, including regions where forced and mutual entrainment compete, with a Bautin point marking the onset of complex cycling behavior. The results provide a detailed phase diagram in the forcing and coupling spaces, with implications for understanding forced synchronization in biological and engineered systems and for exploring how higher-order interactions shape collective dynamics.

Abstract

Synchronization is an important phenomenon in a wide variety of systems comprising interacting oscillatory units, whether natural (like neurons, biochemical reactions, cardiac cells) or artificial (like metronomes, power grids, Josephson junctions). The Kuramoto model provides a simple description of these systems and has been useful in their mathematical exploration. Here we investigate this model combining two common features that have been observed in many systems: external periodic forcing and higher-order interactions among the elements. We show that the combination of these ingredients leads to a very rich bifurcation scenario that produces 11 different asymptotic states of the system, with competition between forced and spontaneous synchronization. We found, in particular, that saddle-node, Hopf and homoclinic manifolds are duplicated in regions of parameter space where the unforced system displays bi-stability.

Figures (5)

• Figure 1: Bifurcation curves in the $(F, \Omega)$ plane for $K_1 = 1$ and $K_{23} = 7$, dividing the plane into ten regions, indicated by capital letters. (a) Bifurcation diagram overview: full lines are Saddle-Node bifurcations (thin black and thick green), dotted lines are SNIPERs and dashed lines are the Hopf curves (red for super-critical and blue for sub-critical). Black squares represent Takens-Bogdanov points, where the Hopf and SN bifurcations meet. (b) Zoom of the thick green SN curve, showing in detail regions E and F. (c) Zoom on the lower Takens-Bogdanov point, with a clear view of the lower homoclinic bifurcation curve (dot-dashed pink line). (d) Zoom on the upper Takens-Bogdanov point, showing the upper homoclinic bifurcation (dot-dashed pink line). The stars mark saddle-node-loop points, where the SNIPER bifurcations turn into SN and the homoclinic bifurcation ends.
• Figure 2: Vector fields, fixed points and periodic orbits of regions A-J shown in Figures \ref{['fig1']}-(a) to (d) for $K_1=1$ and $K_{23}=7$. Full (empty) circles are stable (unstable) nodes and spirals, and crosses are saddle points. Full and dashed lines are stable and unstable orbits, respectively. Specific parameter values for each panel are: (a) $F=0.3$, $\Omega=1.0$; (b) $F=0.3$, $\Omega=0.2$; (c) $F=0.8$, $\Omega=1.0$; (d) $F=1.2$, $\Omega=1.0$; (e) $F=0.17$, $\Omega=0.2$; (f) $F=0.15$, $\Omega=0.1$; (g) $F=0.2612$, $\Omega=0.347$; (h) $F=0.262$, $\Omega=0.3475$; (i) $F=1.4575$, $\Omega=1.495$; (j) $F=1.4555$, $\Omega=1.4895$.
• Figure 3: Saddle-node (solid) and Hopf (dashed) curves for different values of $K_1$ and $K_{23}$. In panels (a)-(c) $K_1=1$ and $K_{23}$ is 5.80, 5.93 and 6.5 respectively. Notice that the thick green saddle-node branch and its correspondent Hopf curve are almost independent of $K_{23}$, whereas the other ones grow as $K_{23}$ increases. In (d)-(e) $K_{23}=7$ and $K_1$ is 1.5 and 2.1 respectively. The thick green branch disappears at $K_1=2$. Black squares show Takens-Bogdanov points. Panel (f) shows the vector field and fixed points found in Region $K$ ($F=0.15$, $\Omega = 0.1$, $K_1=1$, $K_{23}=5.93$).
• Figure 4: Saddle-node and Hopf curves for different values of $F$ and $\Omega$. In panels (a)-(e) $\Omega=0.5$ and $F$ is 0.45, 0.47, 0.49, 0.499 and 0.59, respectively. In panel (f), $\Omega = 0.01$ and $F = 0.02$. Black squares show Takens-Bogdanov points and triangles are Bautin points, where sub and super-critical Hopf bifurcation curves meet (see Fig. \ref{['fig5']}(a)). Regions are labeled according to Figs. \ref{['fig1']}, \ref{['fig2']} and \ref{['fig3']}.
• Figure 5: (a) Zoom of Fig. \ref{['fig4']}(a) showing the joining of sub-critical (blue) and super-critical (red) Hopf bifurcation manifolds at the Bautin point (triangle). A line of saddle-node bifurcation of cycles (purple) starts from the same point. (b) Equilibrium values of the order parameter $r$ as function of $K_1$ for $K_{23}=4$. Solid lines for stable equilibria, dashed lines for unstable ones. Black lines display a pitchfork bifurcation for $F=0$. For $F=0.02$ and $\Omega=0.01$, red lines, the pitchfork unfolds into saddle-node bifurcations. Values of $r<0$ are shown for the sake of visualization.