Emergent dynamics of the Lohe Hermitian sphere model with frustration

TL;DR

This paper investigates the collective behavior of particles on the Hermitian sphere under the Lohe Hermitian Sphere model, revealing conditions for complete and practical aggregation based on coupling and initial states.

Contribution

It introduces a sufficient framework for aggregation in the Lohe Hermitian Sphere model, extending understanding from the classical Lohe sphere model to complex Hermitian settings.

Findings

Complete aggregation occurs under certain conditions for identical ensembles.

Practical aggregation is achieved in non-identical ensembles with specific parameters.

Numerical simulations support the analytical results.

Abstract

We study emergent dynamics of the Lohe hermitian sphere(LHS) model which can be derived from the Lohe tensor model \cite{H-P2} as a complex counterpart of the Lohe sphere(LS) model. The Lohe hermitian sphere model describes aggregate dynamics of point particles on the hermitian sphere $\bbh\bbs^d$ lying in ${\mathbb C}^{d+1}$, and the coupling terms in the LHS model consist of two coupling terms. For 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 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 our analytical results.