Semicontraction and Synchronization of Kuramoto-Sakaguchi Oscillator Networks

Abstract: This paper studies the celebrated Kuramoto-Sakaguchi model of coupled oscillators adopting two recent concepts. First, we consider appropriately-defined subsets of the $n$-torus called winding cells. Second, we analyze the semicontractivity of the model, i.e., the property that the distance between trajectories decreases when measured according to a seminorm. This paper establishes the local semicontractivity of the Kuramoto-Sakaguchi model, which is equivalent to the local contractivity for the reduced model. The reduced model is defined modulo the rotational symmetry. The domains where the system is semicontracting are convex phase-cohesive subsets of winding cells. Our sufficient conditions and estimates of the semicontracting domains are less conservative and more explicit than in previous works. Based on semicontraction on phase-cohesive subsets, we establish the "at most uniqueness" of synchronous states within these domains, thereby characterizing the multistability of this model.