Asymptotic interplay of states and adaptive coupling gains in the Lohe Hermitian sphere model

Junhyeok Byeon; Seung-Yeal Ha; Hansol Park

doi:10.3934/dcdsb.2022007

Article Contents

2022, Volume 27, Issue 11: 6501-6538. Doi: 10.3934/dcdsb.2022007

This issue Previous Article Attractors of the Klein-Gordon-Schrödinger lattice systems with almost periodic nonlinear part Next Article Large-time behavior of solutions to the bipolar quantum Euler-Poisson system with critical time-dependent over-damping

Asymptotic interplay of states and adaptive coupling gains in the Lohe Hermitian sphere model

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
3.
Department of Mathematics, Simon Fraser University, 8888 University Dr, Burnaby, BC V5A 1S6, Canada

^* Corresponding author: Junhyeok Byeon
^* Corresponding author: Junhyeok Byeon

Received: August 31, 2021

Revised: October 31, 2021

Early access: January 2022

Published: October 2022

The work of S.-Y. Ha was supported by National Research Foundation of Korea(NRF-2020R1A2C3A01003881).
The work of H. Park was supported by Pacific Institute for the Mathematical Science(PIMS), Canada postdoctoral fellowship

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Abstract

We study emergent dynamics of the Lohe Hermitian sphere (LHS) model with the same free flows under the dynamic interplay between state evolution and adaptive couplings. The LHS model is a complex counterpart of the Lohe sphere (LS) model on the unit sphere in Euclidean space, and when particles lie in the Euclidean unit sphere embedded in $ \mathbb C^{d+1} $, it reduces to the Lohe sphere model. In the absence of interactions between states and coupling gains, emergent dynamics have been addressed in [23]. In this paper, we further extend earlier results in the aforementioned work to the setting in which the state and coupling gains are dynamically interrelated via two types of coupling laws, namely anti-Hebbian and Hebbian coupling laws. In each case, we present two sufficient frameworks leading to complete aggregation depending on the coupling laws, when the corresponding free flow is the same for all particles.

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Mathematics Subject Classification: Primary: 70G60; Secondary: 34D06.

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