Exact solutions of the harmonically confined Vicsek model

TL;DR

This paper derives exact, stable solutions for the harmonically confined Vicsek model in 2D and 3D, revealing periodic and quasiperiodic swarm formations with quantized confinement, enhancing understanding of collective motion.

Contribution

It provides the first exact solutions of the confined Vicsek model without noise, including stability analysis and diverse orbit configurations.

Findings

Exact periodic and quasiperiodic solutions identified.

Solutions exist at quantized confinement levels.

Stability of solutions confirmed through Floquet analysis.

Abstract

The discrete time Vicsek model confined by a harmonic potential explains many aspects of swarm formation in insects. We have found exact solutions of this model without alignment noise in two or three dimensions. They are periodic or quasiperiodic (invariant circle) solutions with positions on a circular orbit or on several concentric orbits and exist for quantized values of the confinement. There are period 2 and period 4 solutions on a line for a range of confinement strengths and period 4 solutions on a rhombus. These solutions may have polarization one, although there are partially ordered period 4 solutions and totally disordered (zero polarization) period 2 solutions. We have explored the linear stability of the exact solutions in two dimensions using the Floquet theorem and verified the stability assignements by direct numerical simulations.