Emergent behaviors of discrete Lohe aggregation flows

Hyungjun Choi; Seung-Yeal Ha; Hansol Park

doi:10.3934/dcdsb.2021308

Article Contents

2022, Volume 27, Issue 10: 6083-6123. Doi: 10.3934/dcdsb.2021308

This issue Previous Article On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law Next Article Exponential ergodicity for regime-switching diffusion processes in total variation norm

Emergent behaviors of discrete Lohe aggregation flows

1.
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
2.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

^* Corresponding author: Hyungjun Choi
^* Corresponding author: Hyungjun Choi

Received: June 30, 2021

Revised: October 31, 2021

Published: October 2022

The work of S.-Y. Ha was supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881)

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Abstract

The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.

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Mathematics Subject Classification: 82C10, 82C22.

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