Interface geometry controls hysteresis and mode-locking in a simplicial Kuramoto motif

Abstract

Higher-order interactions on simplicial complexes can generate synchronization transitions and multistable collective states that do not appear in purely pairwise network models. Here we study a minimal analytically tractable motif formed by gluing two complete K5 cliques along a common s-clique, with s=1,2,3,4. For a degree-normalized simplicial Kuramoto model with pairwise and triadic interactions, motif symmetries yield an exact low-dimensional reduction on the orbit-synchronous manifold of a symmetric glued motif with orbit-compatible natural frequencies. The resulting effective phase-coupling function contains first and second harmonics whose coefficients are explicit combinatorial functions of the overlap dimension. This gives a direct link between interface geometry and the reduced locking structure. We keep the scope of this statement explicit: the scalar diagonal reduction describes branch existence on x=y, the transverse eigenvalue tests stability only within the two-dimensional reduced dynamics, and the hysteresis observed in the full oscillator system additionally depends on basins of attraction in the full phase space. Within the glued- K5 family and the parameter regimes examined here, changing the overlap dimension modifies the effective harmonic coefficients and thereby broadens or reshapes the principal locking region within the parameter regimes studied. Rotation-number scans show smooth phase slipping outside the principal locking region at the numerical resolution studied here, without ruling out extremely narrow higher-order resonances in other regimes. The analysis clarifies which conclusions follow from the exact symmetry reduction and which require full-system numerical validation.