Emergent behaviors of discrete Lohe aggregation flows

TL;DR

This paper introduces discrete versions of the Lohe sphere and matrix models, analyzing their emergent behaviors and conditions for complete state aggregation using Lyapunov functions and numerical schemes.

Contribution

It proposes novel discrete models for Lohe aggregation flows and provides analytical conditions for their emergent behaviors and state aggregation.

Findings

Discrete Lohe sphere model achieves state aggregation under certain parameters.

Discrete Lohe matrix models exhibit asymptotic state-locking.

Numerical schemes effectively preserve the aggregation properties of the continuous models.

Abstract

The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.