From the Lohe tensor model to the Lohe Hermitian sphere model and emergent dynamics

Abstract

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 analog 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, "$\text{the Lohe tensor model}$." 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 \cite{H-P}. 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.

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