Emergent behaviors of Cucker-Smale flocks on the hyperboloid

TL;DR

This paper extends the study of Cucker-Smale flocking models to hyperbolic spaces, deriving a second-order model on the hyperboloid, analyzing emergent behaviors, and demonstrating two main asymptotic scenarios through theoretical analysis and simulations.

Contribution

It introduces a second-order Cucker-Smale model on the hyperboloid, deriving it from Riemannian geometry, and characterizes the asymptotic flocking behaviors on hyperbolic spaces.

Findings

Velocity alignment is achieved under the model.

Flocking dynamics on b2 exhibit two asymptotic states.

Numerical simulations confirm the theoretical dichotomy.

Abstract

We study emergent behaviors of Cucker-Smale(CS) flocks on the hyperboloid $\mathbb{H}^d$ in any dimensions. In a recent work \cite{H-H-K-K-M}, a first-order aggregation model on the hyperboloid was proposed and its emergent dynamics was analyzed in terms of initial configuration and system parameters. In this paper, we are interested in the second-order modeling of Cucker-Smale flocks on the hyperboloid. For this, we derive our second-order model from the abstract CS model on complete and smooth Riemannian manifolds by explicitly calculating the geodesic and parallel transport. Velocity alignment has been shown by combining general {velocity alignment estimates} for the abstract CS model on manifolds and verifications of a priori estimate of second derivative of energy functional. For the two-dimensional case $\mathbb{H}^2$, similar to the recent result in \cite{A-H-S}, asymptotic flocking admits only two types of asymptotic scenarios, either convergence to a rest state or a state lying on the same plane (coplanar state). We also provide several numerical simulations to illustrate an aforementioned dichotomy on the asymptotic dynamics of the hyperboloid CS model on $\mathbb{H}^2$.