Exact solutions of the harmonically confined Vicsek model

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.

Figures (10)

• Figure 1: (a) Positions of a period-2 exact solution for $\beta=4N$ and initial conditions $z(0)=\frac{1}{2}$, $v(0)=1$. (b) Same for a period-5 solution with $P=5$ in Eq. \ref{['eq3c']} and $z(0)=r_P$, $v(0)=e^{i\pi/P}$. (c) Same for $P=\sqrt{39}$. Colors distinguish positions at different times.
• Figure 2: Positions of a period-4 exact solutions for $\beta=4N$ (line with $z=0,\pm 1$, circle, cross, square and triangle), for $\beta=2N$ [rhombus with $z=\pm i/2,\pm\sqrt{3}/2$, and invariant circle Eq. \ref{['eq4']} with $P=4$]. Shapes and colors distinguish positions belonging to different solutions at different times.
• Figure 3: Partially ordered period-4 solutions on a line. (a) Positions of the particles and (b) positions of the CM for $R_0=2$. (c) Positions of the particles and positions of the CM on the depicted continuous line for $R_0=1$. Here $\beta=1500$ and $N=500$.
• Figure 4: Period-2 solutions for $N=500$. (a) Successive particles positions and (b) successive CM positions after $t=50000$ for a partially ordered exact solution corresponding to $\beta=10^9$, $R_0=2$ and random initial positions inside the unit circle with: $|\mathbf{x}_j(0)|<1$, $\mathbf{x}_j(0)=|\mathbf{x}_j(0)|\mathbf{v}_j(0)$, $\mathbf{v}_j(0)=(\cos\theta,\sin\theta)$, $-\pi<\theta<\pi$. (c) Same as (a) for the totally disordered solution ($W=0$) corresponding to $\beta=2000$, $R_0=1$. The CM position is always at the origin.
• Figure 5: Orbit quantization for a period-12 solution with $\eta=0$, $R_0=0.2$, $G=3$. Successive positions of (a) the particles and (b) the CM for $N=7$, $\beta=4$, $N_1=1$, $N_2=2$, $N_3=4$. The respective periods and radii of the orbits are $P_1=2$, $r_1=1/2$, $P_2=4$, $r_2=1/\sqrt{2}$, and $P_3=6$, $r_3=1$. Initial conditions for the groups: $z_j(0)=r_{j}$, $v_j(0)=ie^{-i\pi/P_j}$, with $j=1,2,3$. (c) Order parameter vs time for the same solution. (d) Stationary period-12 solution obtained by placing 1, 2 and 4 particles at each location of the inner, intermediate and outer orbits of Fig. \ref{['fig5']}(a), respectively. The CM is at the origin, $W=0$, and $N=34$, $\beta=4$.