Emergent dynamics of the inertial Kuramoto model with frustration on a locally coupled graph

TL;DR

This paper investigates how inertial Kuramoto oscillators with frustration synchronize on a network, introducing new energy functions to analyze their dynamics and proving exponential frequency synchronization under certain conditions.

Contribution

It develops a novel analytical framework using convex combinations and energy functions to study synchronization in inertial Kuramoto models with frustration.

Findings

Frequency synchronization occurs exponentially fast.

New energy functions effectively control phase and frequency diameters.

Synchronization results hold under specific initial data and system parameters.

Abstract

We study the synchronized behavior of the inertial Kuramoto oscillators with frustration effect under a symmetric and connected network. Due to the lack of second-order gradient flow structure and singularity of second-order derivative of diameter, we shift to construct convex combinations of oscillators and related new energy functions that can control the phase and frequency diameters. Under sufficient frameworks on initial data and system parameters, we derive first-order dissipative differential inequalities of constructed energy functions. This eventually gives rise to the emergence of frequency synchronization exponentially fast.