Emergent behaviors of the discrete-time Kuramoto model for generic initial configuration

TL;DR

This paper investigates the emergent synchronization behaviors of the discrete-time Kuramoto model with general initial conditions, extending previous work limited to half-circle configurations, and demonstrates exponential synchronization under certain conditions.

Contribution

It develops a discrete gradient flow theory applicable to general Euler schemes and proves exponential synchronization for small mesh sizes with broad initial data.

Findings

Synchronization occurs exponentially fast for small mesh sizes.

The theory applies to general Euler iteration schemes.

Extended understanding beyond half-circle initial configurations.

Abstract

In this paper, we will study the emergent dynamics of the discrete Kuramoto model for generic initial data. This is an extension of the previous work S.-Y. Ha et al. (2019), in which the initial configurations are supposed to be within a half circle. More precisely, we will provide the theory of discrete gradient flow which can be applied to general Euler iteration scheme. Therefore, as a direct application, we conclude the emergence of synchronization of discrete Kuramoto model. Moreover, we obtain for small mesh size that, the synchronization will occur exponentially fast for initial data in A_1 (see definition in (4.1)).