Emergent behaviors of rotation matrix flocks

TL;DR

This paper extends the Cucker-Smale flocking model to the special orthogonal group SO(3), deriving explicit geometric formulas and analyzing emergent velocity alignment through Lyapunov methods, supported by numerical simulations.

Contribution

It provides a novel geometric formulation of the CS model on SO(3) with explicit expressions and analyzes the emergent flocking behavior using Lyapunov and invariance principles.

Findings

Velocity alignment occurs under certain initial conditions.

Characterization of the system's omega-limit set.

Numerical examples confirm analytical results.

Abstract

We derive an explicit form for the Cucker-Smale (CS) model on the special orthogonal group $\mathrm{SO}(3)$ by identifying closed form expressions for geometric quantities such as covariant derivative and parallel transport in exponential coordinates. We study the emergent dynamics of the model by using a Lyapunov functional approach and La Salle's invariance principle. Specifically, we show that velocity alignment emerges from some admissible class of initial data, under suitable assumptions on the communication weight function. We characterize the $\omega$-limit set of the dynamical system and identify a dichotomy in the asymptotic behavior of solutions. Several numerical examples are provided to support the analytical results.