Remarks on the slow relaxation for the fractional Kuramoto model for synchronization

Abstract

The collective behavior of an oscillatory system is ubiquitous in our nature, and one interesting issue in the dynamics of many-body oscillatory systems is the relaxation dynamics toward relative equilibria such as phase-locked states. For the Kuramoto model, relaxation dynamics occurs exponentially fast for generic initial data. However, some synchronization phenomena observed in our nature exhibit a slow subexponential relaxation. Thus, as one of the possible attempts for such slow relaxation, a second-order inertia term was added to the Kuramoto model in the previous literature so that the resulting second-order model can exhibit a slow relaxation dynamics for some range of inertia and coupling strength. In this paper, we present another Kuramoto type model exhibiting a slow algebraic relaxation. More precisely, our proposed model replaces the classical derivative by the Caputo fractional derivative in the original Kuramoto model. For this new model, we present several sufficient frameworks for fractional complete synchronization and practical synchronization.