Open AccessAsymptotic interplay of states and adaptive coupling gains in the Lohe Hermitian sphere model
Author(s) -
Junhyeok Byeon,
SeungYeal Ha,
Hansol Park
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2022007
Subject(s) - hermitian matrix , coupling (piping) , euclidean space , unit sphere , state (computer science) , euclidean geometry , physics , mathematics , mathematical physics , space (punctuation) , unit (ring theory) , quantum mechanics , combinatorics , geometry , computer science , algorithm , mechanical engineering , mathematics education , engineering , operating system
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 \begin{document}$ \mathbb C^{d+1} $\end{document} , 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.