Communications in Mathematical Sciences

Volume 18 (2020)

Number 2

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

Pages: 535 – 570



Xiongtao Zhang (Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, China)

Tingting Zhu (School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China)


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 [Ha et al., J. Math. Phys., 60(5):051508, 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 after some steps for initial data in non-bipolar set and sufficiently large coupling strength.


discrete-time gradient flow, Kuramoto model, discrete-time dynamics, generic initial data, uniform convergence

2010 Mathematics Subject Classification

34D05, 39A10, 39A12, 68M10

The work of X. Zhang is supported by the National Natural Science Foundation of China (Grant No. 1180194).

Received 1 August 2019

Accepted 5 November 2019

Published 20 June 2022