Interplay of inertia and adaptive couplings in the emergent dynamics of Kuramoto ensemble

Abstract

We study the emergent dynamics of Kuramoto oscillators under the interplay between inertia and adaptive couplings. For the phase dynamics, we use the inertial Kuramoto model with time-dependent mutual coupling strengths. For the constant and uniform coupling strength, the inertial Kuramoto model can exhibit the slow relaxation dynamics toward phase-locked state under suitable conditions on system parameters and initial data. To model the time-evolution of mutual coupling strengths, we employ two types of coupling functions, namely “Hebbian coupling” and “anti-Hebbian coupling”. With these modeling spirit, the resulting coupled dynamics for the phase and mutual coupling strength becomes the coupled second-order and first-order systems of ordinary differential equations. For the proposed coupled system, we provide several sufficient frameworks for phase and frequency synchronization in terms of system parameters and initial data for Hebbian and anti-Hebbian coupling functions.