Emergent Dynamics of Kuramoto Oscillators with Adaptive Couplings: Conservation Law and Fast Learning

Abstract

We study an emergent dynamics of the Kuramoto oscillators with adaptive couplings. In the Kuramoto model, pairwise coupling strengths are assumed to be constant and uniform over all interaction pairs. This assumption simplifies the analysis, but it is too restrictive to describe the real applications. In this paper, we relax this uniform strength ansatz by adopting a dynamic feedback law depending on the relative phase differences and discuss two types of adaptive rules for the couplings of oscillators. As a first adaptive model, we consider the adaptive law introduced by Picallo and Riecke and present several sufficient frameworks leading to the asymptotic synchronization. For the second model, we consider Model A introduced in [S.-Y. Ha, S. E. Noh, and J. Park, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 162-194]. We introduce a small parameter that diversifies time scales for the dynamics of the states of the model. By this slow-fast setting, coupling strengths become the fast variables, whereas the phase dynamics becomes the slow one. Tikhonov's theorem guarantees the convergence of a slow-fast dynamical system to a Kuramoto-type model in this singular limit. We also classify admissible phase-locked states for the limit system and provide a sufficient framework leading to the complete phase synchronization in which all oscillators' phases are aggregated to a common phase.