Switching between pairwise and higher-order synchronization on simplicial complexes

Most complex systems, from the brain to ecosystems and multi-agent networks, display pairwise and higher-order interactions that determine their collective behavior. Yet, the regimes in which one interaction order dominate over another, and the mechanisms by which systems move between such regimes remain poorly understood analytically. Here, we consider a network of coupled Stuart-Landau oscillators on time-varying simplicial complexes, for which both pairwise and triplet interactions play a role in phase synchronization. We apply phase reduction and mean-field theory to obtain a reduced model that incorporates structural information through graph and Hodge Laplacians. This results in an analytic formula for a critical coupling surface that delineates regimes of dominance of pairwise- or triplet-mediated synchrony. To quantify how the prevailing synchronization mechanism changes over time, we define an interaction dominance index and investigate how slow modulation of coupling intensities gives rise to regime transitions between pairwise- and triplet-mediated synchrony. Our results provide a tractable framework for understanding how topology and time-dependent coupling jointly control regime switching in systems with coexisting interaction orders.