Emergent dynamics of the Kuramoto ensemble under the effect of inertia

TL;DR

This paper provides analytical proofs for the emergence of complete synchronization in inertial Kuramoto oscillators, extending known results from non-inertial cases and connecting finite systems to mean-field models.

Contribution

It offers new analytical synchronization estimates for inertial Kuramoto systems and establishes the connection to kinetic mean-field models with global solution existence.

Findings

Synchronization occurs in large coupling regimes.

Finite system estimates match mean-field behavior as size grows.

Global existence of measure-valued solutions is proven.

Abstract

We study the emergent collective behaviors for an ensemble of identical Kuramoto oscillators under the effect of inertia. In the absence of inertial effects, it is well known that the generic initial Kuramoto ensemble relaxes to the phase-locked states asymptotically (emergence of complete synchronization) in a large coupling regime. Similarly, even for the presence of inertial effects, similar collective behaviors are observed numerically for generic initial configurations in a large coupling strength regime. However, this phenomenon has not been verified analytically in full generality yet, although there are several partial results in some restricted set of initial configurations. In this paper, we present several improved complete synchronization estimates for the Kuramoto ensemble with inertia in two frameworks for a finite system. Our improved frameworks describe the emergence of phase-locked states and its structure. Additionally, we show that as the number of oscillators tends to infinity, the Kuramoto ensemble with infinite size can be approximated by the corresponding kinetic mean-field model uniformly in time. Moreover, we also establish the global existence of measure-valued solutions for the Kuramoto equation and its large-time asymptotics.