Multi-Population Phase Oscillator Networks with Higher-Order Interactions

Abstract: The classical Kuramoto model consists of finitely many pairwise coupled oscillators on the circle. In many applications a simple pairwise coupling is not sufficient to describe real-world phenomena as higher-order (or group) interactions take place. Hence, we replace the classical coupling law with a very general coupling function involving higher-order terms. Furthermore, we allow for multiple populations of oscillators interacting with each other through a very general law. In our analysis, we focus on the characteristic system and the mean-field limit of this generalized class of Kuramoto models. While there are several works studying particular aspects of our program, we propose a general framework to work with all three aspects (higher-order, multi-population, and mean-field) simultaneously. Assuming identical oscillators in each population, we derive equations for the evolution of oscillator populations in the mean-field limit. First, we clarify existence and uniqueness of our set of characteristic equations, which are formulated in the space of probability measures together with the bounded-Lipschitz metric. Then, we investigate dynamical properties within the framework of the characteristic system. We identify invariant subspaces and stability of the state, in which all oscillators are synchronized within each population. Even though it turns out that this so called all-synchronized state is never asymptotically stable, under some conditions and with a suitable definition of stability, the all-synchronized state can be proven to be at least locally stable. In summary, our work provides a rigorous mathematical framework upon which the further study of multi-population higher-order coupled particle systems can be based.