Phase locking and multistability in the topological Kuramoto model on cell complexes

TL;DR

This work extends the Kuramoto framework to topological (higher-order) interactions on cell complexes and introduces a nonlinear Kirchhoff-like algorithm to identify all phase-locked states. By applying the method to rings, Platonic solids, and simplexes, it reveals that multistability emerges when boundary structures involve more than four cells, producing structural cascades of stable states across dimensions and universality patterns tied to boundary topology. The approach unifies linear and nonlinear effects, offering a constructive procedure to enumerate phase-locked solutions and bound winding numbers, with implications for neuronal, optical, and power-grid systems where higher-order couplings are intrinsic. The findings emphasize the richness of dynamics enabled by higher-order connectivity and provide a versatile framework for analyzing diffusion, consensus, and pattern formation on cell complexes.

Abstract

The topological Kuramoto model generalizes classical synchronization models by including higher-order interactions, with oscillator dynamics defined on cells of arbitrary dimension within simplicial or cell complexes. In this article, we demonstrate multistability in the topological Kuramoto model and develop the topological nonlinear Kirchhoff conditions algorithm to identify all phase-locked states on arbitrary cell complexes. The algorithm is based on a generalization of Kirchhoff's laws to cell complexes of arbitrary dimension and nonlinear interactions between cells. By applying this framework to rings, Platonic solids, and simplexes, as minimal representative motifs of larger networks, we derive explicit bounds (based on winding number constraints) that determine the number of coexisting stable states. We uncover structural cascades of multistability, inherited from both lower and higher dimensions and demonstrate that cell complexes can generate richer multistability patterns than simplicial complexes of the same dimension. Moreover, we find that multistability patterns in cell complexes appear to be determined by the number of boundary cells, hinting a possible universal pattern.

Figures (8)

• Figure 1: From Kirchhoff’s circuit law to the topological nonlinear Kirchhoff conditions. Kirchhoff’s voltage law enforces linear conservation of potential differences around each circuit loop. In the nonlinear generalization, phase differences $\psi_i$ across network edges satisfy a nonlinear constraint, $\sum_i \arcsin(\psi_i) = 2\pi z$, where integer winding numbers $z$ describe distinct phase-locked states. The topological generalization extends this principle to higher-order interactions on cell complexes, where linear combinations of nonlinear phase differences across higher-dimensional cells satisfy $\sum_i C_i \arcsin \psi_i^{[\pm]} = 2\pi z_i^{[\pm]}$, with $C_i$ determined by the boundary structure of each cell. Here, the dynamics is defined on edges, and interactions occur via nodes in the lower dimension ($\psi_i^{[-]}$) and faces in the upper dimension ($\psi_i^{[+]}$).
• Figure 2: Examples of cell complexes for $n=1$. Two representative cell complexes, (a) a ring and (b) a cube, where the dynamics is defined on the edges, and interactions occur via both nodes and faces. Nodes are shown in black, edges in blue and faces in light green.
• Figure 3: Phase locked solutions of the topological Kuramoto model for a six-edge ring. Each marker represents a distinct phase-locked solution obtained from the nonlinear Kirchhoff conditions. Filled (empty) markers indicate linearly stable (unstable) solutions. Every partition except $|S_{[0]}^{\bullet}|=|S_{[0]}^{\circ}|=3$ gives rise to phase-locked solutions. For $|S_{[0]}^{\bullet}|=6, |S_{[0]}^{\circ}|=0$, three distinct stable phase-locked solutions coexist, illustrating multistability in the minimal ring motif.
• Figure 4: Effect of asymmetry in natural frequencies on stable phase-locked solutions in the ring with six edges. Top row: Plots of condition \ref{['eq:KVL2']} for two node partitions, $S_{[0]}^\circ = \emptyset$ (solid lines) and $S_{[0]}^\circ = [5, 6]$ (dashed lines) at selected values of $\omega_0$ in ${\boldsymbol{\omega}} = (0, 0, 0, 0, \omega_0, -\omega_0)$. Filled (empty) symbols mark stable (unstable) solutions. The number of intersections with integer values decreases as $\omega_0$ increases, i.e. phase locked states gradually disappear. The inset highlights a narrow interval of $\omega_0$ where $S_{[0]}^\circ = [5, 6]$ crosses zero twice, producing one stable and one unstable solution, while $S_{[0]}^\circ = \emptyset$ does not cross it. Bottom row: Dependence of the corresponding phase-locked states on $\omega_0$. The stable solution $S_{[0]}^\circ = \emptyset$ (orange line) vanishes near $\omega_0 \approx 1.309$, giving rise to the stable solution $S_{[0]}^\circ = [5, 6]$ (solid blue line). Stable and unstable (dashed blue line) solutions of $S_{[0]}^\circ = [5, 6]$ annihilate in a saddle-node bifurcation near $\omega_0 \approx 1.327$.
• Figure 5: Dependence of the number of stable phase-locked states on ring size. The number of stable solutions $N_{stable}$ as a function of ring size $s$ in an all-normal partitioned ring. Symbols show numerical results, and the solid line denotes the analytical prediction from Eq. \ref{['eq:z_ring0']}.