Figure 1

(Color online) Snapshots of the configuration in position space for the model given in Eqs. (1, 2), with $\kappa =1$ and $t_c=10$, for systems of $N=100$ particles. The small arrows represent the velocity vectors ${\mathbf{v}}_i$ for each one of them while the big (red) arrows, across the figures, correspond to the way of increasing noise intensity. In (a) and (b), and (c) and (f), the translational and oscillatory states are, respectively, shown in regions I and IV for different connectivity values (see text for more details). When the system shows intermittent dynamics, (d) a laminar regime can be interrupted by (e) a turbulence burst. In (d) and (e), the value of $\eta$ considered was taken inside region II (see text).Reuse & Permissions

Figure 2

(Color online) Stationary order parameter $\Psi$, given in Eq. (8a), as a function of the noise intensity $\eta$. Equations (1, 2) were numerically solved for systems of $N=200$ particles with $\kappa =1$ and $t_c=10$. Initial conditions were taken in the TranS (a) and in the OscS (b). The changes of stability and the different kinds of phase transitions are apparent as the connectivity $\mu /N$ changes. See text for more detailsReuse & Permissions

Figure 3

(Color online) Times series of the order parameter $\Psi \left(t\right)=\left|{\mathbf{V}}_{CM}\left(t\right)\right|/v_0$ for different values $\mu /N$ and $\eta$. Typical cases inside region III for different connectivities are shown in (d), where the curves are vertically displaced for better clarity. Note that the time intervals between turbulence bursts, and between free transitions to the translational and oscillatory states inside region III, are in general larger than the time scale of the noise-induced fluctuations for $\mu /N\le 0.2$. See the text for more details.Reuse & Permissions

Figure 4

(Color online) Stationary probability distribution functions of the order parameter $\Psi$ for selected cases of Fig. 2 with finite $\mu /N$. In (a)–(c), the filled curves correspond to values of $\eta$ inside region III, where the system shows bistability and free transitions between the TranS and the OscS through the bimodal distribution of $P\left(\Psi \right)$. This behavior is typical of first-order phase transitions but only close to the critical point. In (d), the filled curve denotes the unimodal character of $P\left(\Psi \right)$ in the transition from the TranS to the OscS, typical of second-order phase transitions.Reuse & Permissions

Figure 5

(Color online) Stationary angular momentum $\Lambda$, given in Eq. (8b), as a function of $\eta$ with $N=200$, $\kappa =1$, and $t_c=10$, for initial conditions taken in the TranS (a) and in the OscS (b). For the case $\mu /N=0$, the vertical dotted line corresponds to the critical point ${\eta }_c$ of the phase transition in $\Psi$ from the TranS to the OscS when initial conditions are taken in the TranS. Beyond that line the only solution of the system is the OscS regardless of initial conditions. On the other hand, beyond the vertical dash-dot-dashed line, the correlated angular noise is strong enough to randomly change the direction of rotation of the particles in the OscS. In contrast, in between the two vertical lines and after the system reaches the stationary OscS, particles rotating in either direction, clockwise or counterclockwise, will remain rotating in such a way regardless of the fluctuations induced by the noise. This is more evident in (a) where $\Lambda$ presents some bumps product of an asymmetric number of particles rotating in both directions, due to the transition the system undergoes from the TranS to the OscS. See text for more details on the cases with finite connectivities.Reuse & Permissions

Figure 6

(Color online) Probability distribution functions of the magnitude of the angular momentum of the individual particles $L_i$ for systems with $N=200$. In (a), for $\mu /N=0$, the two peaks away from the center for the curve with $\eta =0.564$ indicate a limit-cycle oscillatory state. That is also the case in (b), for the different values of $\mu /N$ with a one-peaked $P_{\lambda }\left(L\right)$ away from the center. In (c) and (d), the filled curves correspond to values of $\eta$ inside region III. In (c) the tails in some of the distributions indicate the development of limit-cycle-like oscillatory states. In contrast, in (d), the tails of the distributions decay faster from the center indicating the development of qualitatively disorder oscillatory states.Reuse & Permissions

Figure 7

(Color online) Above, (a)–(c), mean-square dispersions, given in Eqs. (9a, 9b), and the mean range of the alignment interaction, $⟨r_0⟩$ (see text for definition), as a function of $\eta$, for different connectivities, $N=200$ and $t_c=10$. The curves with dashed lines and solid symbols correspond to numerical results with initial condition in the OscS, while the curves with solid lines and clear symbols correspond to numerical results with initial condition in the TranS. The different stability regions are identified with roman numerals for the different cases. Below, behavior of the maximum transversal dispersion, ${\Sigma }_{\perp }^{\text{MAX}}$, as a function of (d) $t_c$ and (e) $\mu /N$.Reuse & Permissions

Figure 8

(Color online) Snapshots of the configuration in position space for the model given in Eqs. (1, 2) with $\kappa =1$, $t_c=10$, and $N=200$. The small arrows represent the velocity vectors ${\mathbf{v}}_i$ for each one of the particles. These states were obtained from simulations with initial conditions in the OscS. In (b) the large arrow points to the trajectory of the center of mass (solid line) for a finite period of time.Reuse & Permissions

Figure 9

Plot $\mu /N$ vs $N$ where the solid line with grey circles corresponds to the crossover regime that separates the cases where the system exhibits a bistable region (region III) with free transitions between the TranS and the Oscs (depicted with clear circles), from those where the transition from the TranS to the OscS is apparently continuous (depicted with solid circles). In all of the cases $t_c=10$ and $\kappa =1$.Reuse & Permissions

Figure 10

(Color online) [(a)–(d)] Semi-log plots of the PDF of $\Psi$ with $N=200$, $t_c=10$, and $\kappa =1$, for different values $\mu /N$ and $\eta$. In (a), the selected value of $\eta$ lies in the subcritical region of the phase transition from the TranS to the OscS. In (b)–(d), the selected values of $\eta$ lie inside region II. The dotted lines correspond to Gaussian fits of the whole distribution in (a), and of the crest of the distributions in (b)–(d).Reuse & Permissions

Figure 11

(Color online) Normalized magnitude of the mean velocity of the group, in the steady state, given by the order parameter $\Psi$ of Eq. (A5), for the Vicsek et al. model, as a function of the noise intensity $\eta$ while the limit $N\to \infty$ is approached. Numerical (curves with clear symbols) and analytical (dashed lines) results, the latter corresponding to the expression of Eq. (A8) for different values of $N$, are presented assuming a global alignment interaction and a constant density $\rho ={10}^3$. On the other hand, when a local alignment interaction is assumed, numerical results (curves with solid symbols) are presented as $N\to \infty$ by making $\rho \to \infty$ for a fixed $L=10$. The black solid curve corresponds to the analytical approximation of Eq. (A9). In all the simulations for the Vicsek et al. model only the small velocity regime [26] was considered $\left(v_0=0.03\right)$ with $r=1$. The inset shows numerical results (grey circles) for model (1) with $N=200$, $k=1$, and $t_c=10$, when a global alignment interaction $\left(\mu /N=N-1\right)$ is considered, against the analytical exact result (black solid curve) of Eq. (A4).Reuse & Permissions