Cucker-Smale type flocking models on a sphere

TL;DR

This paper introduces a novel Cucker-Smale flocking model on a sphere, incorporating new geometric and interaction terms, and proves velocity alignment and flocking behavior under certain conditions.

Contribution

The paper develops a Cucker-Smale type model on a sphere with new terms for centripetal force and bonding, and proves flocking and velocity alignment results.

Findings

Velocity alignment is proven for the model.

Flocking occurs with large bonding forces.

Antipodal point avoidance is demonstrated.

Abstract

We present a Cucker-Smale (C-S) type flocking model on a sphere. We study velocity alignment on a sphere and prove the emergence of flocking for the proposed model. Our model includes three new terms: a centripetal force, multi-agent interactions on a sphere and inter-particle bonding forces. To compare velocity vectors on different tangent spaces, we introduce a rotation operator in our new interaction term. Due to the geometric restriction, the rotation operator is singular at antipodal points and the relative velocity between two agents located at these points is not well-defined. Based on an energy dissipation property of our model and a variation of Barbalat's lemma, we show the alignment of velocities for an admissible class of communication weight functions. In addition, for sufficiently large bonding forces we prove time-asymptotic flocking which includes the avoidance of antipodal points.