Exact solutions of the harmonically confined Vicsek model Articles uri icon

publication date

  • November 2024

start page

  • 1

end page

  • 10

issue

  • 115826

volume

  • 191

International Standard Serial Number (ISSN)

  • 0960-0779

Electronic International Standard Serial Number (EISSN)

  • 1873-2887

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.

subjects

  • Mathematics

keywords

  • vicsek model; flocking; exact periodic orbits; invariant circles; floquet theorem; harmonic confinement; orbit quantization