Self-propelled chimeras

The synchronization of self-propelled particles (SPPs) is a fascinating instance of emergent behavior in living and manmade systems, such as colonies of bacteria, flocks of birds, robot ensembles, and many others. The recent discovery of chimera states in coupled oscillators opens up new perspectives and indicates that other emergent behaviors may exist for SPPs. Indeed, for a minimal extension of the classical Vicsek model we show the existence of chimera states for SPPs, i.e., one group of particles synchronizes while others wander around chaotically. Compared to chimeras in coupled oscillators where the site position is fixed, SPPs give rise to new distinctive forms of chimeric behavior. We emphasize that the found behavior is directly implied by the structure of the deterministic equation of motion and is not caused by exogenous stochastic excitation. In the scaling limit of infinitely many particles, we show that the chimeric state persists. Our findings provide the starting point for the search or elicitation of chimeric states in real world SPP systems.

Figure 1

Phase diagram for model Eq. (1) in the $(\alpha ,\rho )$ parameter plane. Localized (LC) and nonlocalized (NLC) chimeras exist in yellow (light gray) and blue (dark gray) regions, respectively. Snapshots demonstrate typical chimera regimes, i.e., A*=(1.54,0.3), B*=(1.5,0.03), respectively. Particles are colored with respect to averaged phase velocity $〈\dot{\phi }〉$ (cf. Figs. 2 and 2; averaging time is $t=5$ time units; see videos S1 and S2 in Refs. [29, 30]) subject to binary thresholding. Intermittent behavior is characteristic for the neighboring region (oblique hatching). The region to the left leads to complete phase synchronization (SYNC) and the region to the right leads to full disorder (CHAOS). Lines A1-A2 and B1-B2 are used in Fig. 5 for order parameter description. Other parameters are $\sigma =1.0,N=1000$.

Figure 2

Emergence of chimeras. (a) Space-time plot for LC, based on the averaged phase velocity. Along with it, profiles of averaged phase velocity $〈\dot{\phi }〉$ and averaged phase $〈\phi 〉$ are provided for time points from characteristic periods of formation. (b) Space-time plot for LC, based on the averaged value of distance from a particle position to a point of maximum density. Along with it, the corresponding profiles are provided at the same times as in (a). (c) Space-time plot for NLC in the same manner as in (a). Particles are reordered identically at every time step with respect to the value of averaged angular velocity from (a) or (c), (b) is ordered according to (a). Temporal averaging is $t=10$. Time points for (a) and (b) are $t_1=50$, $t_2=255$, and $t_3=1500$; time points for (c) are $t_1=50$, $t_2=210$, and $t_3=1500$. Parameters: (a, b) $\sigma =1.0,\rho =0.3,\alpha =1.54$; (c) $\sigma =1.0,\rho =0.03,\alpha =1.5$.

Figure 3

Localization traits of LC. (a) Traces of all particles over time ($t=100$). Light circular trajectories indicate the motion of particles from a synchronized group whilst the dark background of considerable nonzero mass indicates the uniform spreading of the rest of chaotic particles. The color code corresponds to the number of particles at a particular position, normalized by the number of particles $N$ and by the averaging time. Coupling strength is $\sigma =4$ to keep the whole circular trajectory inside a simulation box. (b) Image of trajectories for one particle from a localized cluster (black circle in the middle) and for five different particles from the chaotic cluster (here $\sigma =1$) for unrolled periodic boundary conditions. The tracking time is $t=100$. (c) Radius of rotation $\xi$ for particles from a localized group as a function of the coupling strength $\sigma$. (d) Number of particles, which leave and join the synchronized cluster, per cluster cycle ($\approx 10$ time units) per population size. Other parameters are $\rho =0.3,\alpha =1.54,N=1000$.

Figure 4

Local Lyapunov exponents. (a) Local Lyapunov spectra ${\mathrm{\Lambda }}_x$, ${\mathrm{\Lambda }}_y$, and ${\mathrm{\Lambda }}_{\phi }$ in the LC case for all particles for $x$, $y$, $\phi$ variables, respectively. The exponents were computed after 100 time units after the chimera has been formed. Particles are ordered with respect to $〈\dot{\phi }〉$ in the same manner as in Fig. 2. Other parameters are $\sigma =1.0,\rho =0.3,\alpha =1.54$. (b) Local Lyapunov spectra ${\mathrm{\Lambda }}_x$, ${\mathrm{\Lambda }}_y$, and ${\mathrm{\Lambda }}_{\phi }$ in the NLC case for all particles for $x,y,\phi$ variables, respectively. The exponents were computed after 10 time units after the chimera has been formed. Particles are ordered with respect to $〈\dot{\phi }〉$ in the same manner as in Fig. 2. Other parameters are $\sigma =1.0,\rho =0.03,\alpha =1.5$.

Figure 5

Summary statistics for chimera states. Evolution of the order parameter magnitude $R$ over time for (a) LC and (b) NLC. Temporal averaging of $t=10$ is applied. Insets indicate changes in the synchronization level along the lines A1-A2 and B1-B2 from Fig. 1. (c) The time of maximum synchronization quantified through $R$ versus phase lag $\alpha$, for different radii $\rho \in \{0.1,0.3,0.5\}$. (d),(e) Pair distribution functions for LC and NLC, respectively. Other parameters are the same as in Fig. 2.

Figure 6

Local complex mean field $Z_k\left(t\right)=R_k\left(t\right)exp\left[i{\mathrm{\Theta }}_k\left(t\right)\right]$ at $t=1000$ for (a) LC and (b) NLC. The ordering of particles is the same as in Figs. 2 and 2, respectively. Parameters: (a) $\sigma =1.0,\rho =0.3,\alpha =1.54$; (b) $\sigma =1.0,\rho =0.03,\alpha =1.5$.

Figure 7

Summary statistics for chimera states under noise. (a) Values of global order parameter magnitude $R$ versus noise intensity $D_{\phi }$ for solutions of Eq. (3). Different colors (line styles) correspond to different coupling coefficients $\sigma$; (b) localization measure $H$ versus noise intensity $D_{\phi }$. A shaded region shows the standard deviation along a curve. Other parameters are $\rho =0.3,\alpha =1.54,N=1000$.

Figure 8

Chimera states due to the generalized model Eq. (4) with the cosine kernel Eq. (5). (a) Localized chimera state with parameters $\sigma =1.0,\alpha =1.53,A=1.0$; (b) nonlocalized chimera state with parameters $\sigma =1.0,\alpha =1.56,A=1.0$.

Figure 9

Chimera states due to the generalized model Eq. (4) with the exponential kernel Eq. (6). (a) Localized chimera state with parameters $\sigma =1.0,\alpha =1.54,k=4.0$; (b) nonlocalized chimera state with parameters $\sigma =1.0,\alpha =1.54,k=20.0$.

Figure 10

Solution for the 3+1-dimensional density function from the continuum limit Eq. (8), projected into spatial coordinates $(x,y)$. The projection was done so as to ensure $f(x,y,t)={\int }_0^{2\pi }f(x,y,\phi ,t)d\phi$. The color corresponds to the density value, normalized per each frame, so that all dynamical variations are captured. The complete dynamics of $f(x,y,\phi ,t)$ and the original density scale are provided in Fig. 12. The solution here is shown as a sequence of characteristic phases that the system passes (see their description in the text).

Figure 11

Comparison of the dynamics in the case of the nonlocalized chimera state from different models: (i) yellow histogram of partiles' directions is given by the solution of a finite-size particle model Eq. (3) with the number of particles $N=5\times {10}^4$; (ii) solid blue curve represents the projection of the solution for a 3+1-dimensional partial differential Eq. (8) into $\phi$ coordinates; (iii) dashed red curve represents the solution of a reduced partial differential Eq. (9). Parameters for the three models are $\sigma =1.0,\rho =0.03,\alpha =1.5,D_{\phi }=0.01$.

Figure 12

Solution for the 3+1-dimensional one-particle density function from the continuum limit Eq. (C8) in the regime of localized chimera with parameters $\sigma =4.0,\rho =0.3,\alpha =1.54,D_{\phi }=0.01$. Color code corresponds to the density value, rescaled at each time point. Different snapshots represent the solution of the system at characteristic phases: (a) initial density values are assigned a constant value plus some perturbation; (b) most of the mass gathers in patches of high density; (c) the patches merge in tubes, which in turn begin to aggregate; (d) most of the mass gathers in a layer that is uniform in $(x,y)$; (e) the layer starts to shrink and wavy structures appear in front of it; (f) the layer compresses considerably and the waves diminish; (g) the layer becomes inhomogeneous in $(x,y)$; (h) the elongated structure of high density emerges; (i) the final high-density cluster of ellipsoidal shape forms and moves along the helical trajectory.