均匀Lohe厄米球粒子在时滞相互作用下的涌现行为
Emergent behaviors of homogeneous Lohe Hermitian sphere particles under time-delayed interactions
摘要：
研究了具有时滞的Lohe厄米球（LHS）模型在齐次系综下的突现行为。LHS模型是欧氏空间中单位球上Lohe球（LS）聚集模型的复对应，它用$d \geq 2$描述了$\mathbb{C}^d$中单位厄米球上粒子的聚集，最近它被两位作者作为Lohe张量模型的特例引入[23]。当耦合增益对满足特定的线性关系，即Stuart-Landau（SL）耦合增益对时，可以将其嵌入到$\mathbb{R}^{2d}$上的LS模型中。在这项工作中，我们表明，如果耦合增益对接近SL耦合对的情况下，LHS模型的动力学表现出一个新兴的聚集现象，通过时滞之间的相互作用和状态之间的非线性耦合。为此，我们提出了几个框架，完全聚合和实际聚合的初始数据和系统参数使用李雅普诺夫函数的方法。
英文摘要：
We study emergent behaviors of the Lohe hermitian sphere(LHS) model with a time-delay for a homogeneous 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 [23]. 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.
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