We study emergent dynamics of the Lohe Hermitian sphere (LHS) model, which can be derived from the Lohe tensor model [S.-Y. Ha and H. Park, SIAM J. Appl. Dyn. Syst. 13, 1312–1342 (2020)] as a complex counterpart of the Lohe sphere model. The LHS model describes aggregate dynamics of point particles on the Hermitian sphere ${\mathbb{H}\mathbb{S}}^d$ lying in ${\mathbb{C}}^{d+1}$, and the coupling terms in the LHS model consist of two terms. For an identical ensemble with the same free flow dynamics, we provide a sufficient framework, leading to the complete aggregation in which all point particles form a giant one-point cluster asymptotically. In contrast, for a non-identical ensemble, we also provide a sufficient framework for the practical aggregation. Our sufficient framework is formulated in terms of coupling strengths and initial data. We also provide several numerical examples and compare them with analytical results.

中文翻译：

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沮丧的Lohe Hermitian球面模型的新兴动力学

我们研究了 Lohe Hermitian 球 (LHS) 模型的涌现动力学，该模型可以从 Lohe 张量模型 [S.-Y. Ha 和 H. Park，SIAM J. Appl。达因。系统。13 , 1312–1342 (2020)] 作为 Lohe 球体模型的复杂对应物。LHS 模型描述了 Hermitian 球体上点粒子的聚合动力学${\mathbb{H}\mathbb{秒}}^d$ 躺在 ${\mathbb{C}}^{d+1}$，LHS 模型中的耦合项由两项组成。对于具有相同自由流动动力学的相同系综，我们提供了一个足够的框架，导致所有点粒子渐近地形成一个巨大的单点簇的完全聚集。相比之下，对于不相同的集成，我们还为实际聚合提供了足够的框架。我们充分的框架是根据耦合强度和初始数据制定的。我们还提供了几个数值例子并将它们与分析结果进行了比较。