Emergent dynamics of the Kuramoto ensemble under the effect of inertia

Young-Pil Choi,Seung-Yeal Ha,Javier Morales

doi:10.3934/dcds.2018213

Young-Pil Choi, Seung-Yeal Ha + Show 1 more

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https://doi.org/10.3934/dcds.2018213

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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.

Highlights

Collective behaviors of complex system is one of the important characteristics that are often observed in classical and quantum many-body systems, e.g., synchronous firing of flash, swarming of fish and flocking of birds, array of Josephson junctions, etc [1, 2, 3, 8, 26, 36, 38, 45]
We focus on a generalized Kuramoto model with inertia which was introduced in [23] to explain the slow relaxation of firefly Pteroptyx malaccae’s rhythmn
We addressed the complete synchronization problem for the particle and kinetic Kuramoto models for identical oscillators with homogeneous and heterogenous inertia and friction

Summary

Introduction

Collective behaviors of complex system is one of the important characteristics that are often observed in classical and quantum many-body systems, e.g., synchronous firing of flash, swarming of fish and flocking of birds, array of Josephson junctions, etc [1, 2, 3, 8, 26, 36, 38, 45]. Among such diverse collective behaviors, our main interest in this paper lies on the synchronization phenomenon, roughly speaking “adjustment of rhythms in weakly coupled oscillators due to weak interactions”, i.e., oscillators adjust their rhythms and exhibit a common rhythm like a single giant oscillators. Let θi = θi(t) be the phase of the Kuramoto oscillator at vertex i, and we denote mi by the strength of inertia (mass) of the i-th oscillator In this setting, the temporal dynamics of the phase θi is given by the following second-order system: N miθi = −γiθi + νi + κ aij sin(θj − θi), j=1

Objectives

Conclusion