Connecting Hodge and Sakaguchi-Kuramoto through a mathematical framework for coupled oscillators on simplicial complexes

Phase synchronizations in models of coupled oscillators such as the Kuramoto model have been widely studied with pairwise couplings on arbitrary topologies, showing many unexpected dynamical behaviors. Here, based on a recent formulation the Kuramoto model on weighted simplicial complexes with phases supported on simplices of any order k, we introduce linear and non-linear frustration terms independent of the orientation of the k + 1 simplices, as a natural generalization of the Sakaguchi-Kuramoto model to simplicial complexes. With increasingly complex simplicial complexes, we study the the dynamics of the edge simplicial Sakaguchi-Kuramoto model with nonlinear frustration to highlight the complexity of emerging dynamical behaviors. We discover various dynamical phenomena, such as the partial loss of synchronization in subspaces aligned with the Hodge subspaces and the emergence of simplicial phase re-locking in regimes of high frustration. Synchronization dynamics in the presence of higher order interactions is well represented through variations of the Kuramoto model and subject of current interest. Here, the authors study and characterize the behavior of the simplicial Kuramoto model with weights on any simplices and in the presence of linear and nonlinear frustration, defined as the simplicial Sakaguchi-Kuramoto model.