SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1312-1342, January 2020.
We study emergent behaviors of the Lohe hermitian sphere (LHS) model which is an aggregation model on ${\mathbb C}^d$. The LHS model is a complex analogue of the Lohe sphere model on ${\mathbb R}^d$, and hermitian spheres are invariant sets for the LHS dynamics. For the derivation of the LHS model, we use a top-down approach, namely a reduction from a high-rank aggregation model, the Lohe tensor model. The Lohe tensor model is a first-order aggregation model on the space of tensors with the same rank and sizes, and it was first proposed by the authors in a recent work [J. Stat. Phys., 178 (2020), pp. 1268--1292]. In this work, we study how the LHS model appears as a special case of the Lohe tensor model, and for the proposed model, we provide a cross-ratio-like conserved quantity, a sufficient framework for the complete aggregation, and a uniform $\ell^p$-stability estimate with respect to initial data.

中文翻译：

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从Lohe张量模型到Lohe Hermitian球体模型和新兴动力学

SIAM应用动力系统杂志，第19卷，第2期，第1312-1342页，2020年1月。
我们研究了Lohe Hermitian球（LHS）模型的新兴行为，该模型是基于$ {\ mathbb C} ^ d $的聚集模型。LHS模型是$ {\ mathbb R} ^ d $上Lohe球体模型的复杂类似物，而Hermitian球体是LHS动力学的不变集。对于LHS模型的推导，我们使用自上而下的方法，即从高级聚合模型Lohe张量模型进行简化。Lohe张量模型是在具有相同秩和大小的张量的空间上的一阶聚合模型，它是由作者在最近的工作中首次提出的。统计 Phys。178（2020），第1268--1292页]。在这项工作中，我们研究了LHS模型是如何作为Lohe张量模型的特例出现的，对于提出的模型，我们提供了一个类似跨比率的守恒量，这是完整聚合的充分框架，