Exact solutions of the harmonically confined Vicsek model

L.L. Bonilla and R. González-Albaladejo

Chaos, Solitons & Fractals, 2025, vol. 191, issue C

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 assignments by direct numerical simulations.

Keywords: Vicsek model; Flocking; Exact periodic orbits; Invariant circles; Floquet theorem; Harmonic confinement; Orbit quantization (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:191:y:2025:i:c:s096007792401378x

DOI: 10.1016/j.chaos.2024.115826

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