Communications in Mathematical Sciences

Volume 21 (2023)

Number 6

Discrete perturbed gradient flow and its application

Pages: 1505 – 1530

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n6.a3

Authors

Lingzhi Hao (School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China)

Xiongtao Zhang (School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China)

Abstract

We study discrete dynamical system with perturbed gradient flow structure and its related applications. We prove that states with uniform bound will eventually converge to an equilibrium state, where Łojasiewicz inequality plays an important role. Moreover, the convergence rate is uniform with respect to the mesh size, which implies uniform transition from discrete time model to continuous time model. As direct applications, we use this theory to prove the emergent dynamics in discrete thermodynamic Kuramoto model and swarmalator model.

Keywords

discrete perturbed gradient flow, Łojasiewicz inequality, discrete swarmalator model

2010 Mathematics Subject Classification

34D05, 39A10, 39A12, 68M10

Received 2 November 2021

Received revised 14 October 2022

Accepted 15 November 2022

Published 22 September 2023