# American Institute of Mathematical Sciences

September  2021, 16(3): 459-492. doi: 10.3934/nhm.2021013

## Emergent behaviors of Lohe Hermitian sphere particles under time-delayed interactions

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 and, Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Republic of Korea 2 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Gyuyoung Hwang

Received  January 2021 Revised  April 2021 Published  September 2021 Early access  May 2021

Fund Project: The work of S.-Y. Ha was supported by National Research Foundation of Korea(NRF-2020R1A2C3A01003881). The work of H. Park was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585)

We study emergent behaviors of the Lohe Hermitian sphere(LHS) model with a time-delay for a homogeneous and heterogeneous ensemble. The LHS model is a complex counterpart of the Lohe sphere(LS) aggregation model on the unit sphere in Euclidean space, and it describes the aggregation of particles on the unit Hermitian sphere in $\mathbb C^d$ with $d \geq 2$. Recently it has been introduced by two authors of this work as a special case of the Lohe tensor model. When the coupling gain pair satisfies a specific linear relation, namely the Stuart-Landau(SL) coupling gain pair, it can be embedded into the LS model on $\mathbb R^{2d}$. In this work, we show that if the coupling gain pair is close to the SL coupling pair case, the dynamics of the LHS model exhibits an emergent aggregate phenomenon via the interplay between time-delayed interactions and nonlinear coupling between states. For this, we present several frameworks for complete aggregation and practical aggregation in terms of initial data and system parameters using the Lyapunov functional approach.

Citation: Seung-Yeal Ha, Gyuyoung Hwang, Hansol Park. Emergent behaviors of Lohe Hermitian sphere particles under time-delayed interactions. Networks & Heterogeneous Media, 2021, 16 (3) : 459-492. doi: 10.3934/nhm.2021013
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##### References:
Exponential aggregation for $\tau>0$, $N = 4$ and $d = 2$
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