Synchronization in Dynamical Networks of Locally Coupled Self-Propelled Oscillators

The emergent cooperative behavior of mobile physical entities exchanging information with their neighborhood has become an important problem across disciplines, thus requiring a general framework to describe such a variety of situations. We introduce a generic model to tackle this problem by considering the synchronization in time-evolving networks generated by the stochastic motion of self-propelled physical interacting units. This framework generalizes previous approaches and brings a unified picture to understand the role played by the network topology, the motion of the agents, and their mutual interaction. This allows us to identify different dynamic regimes where synchronization can be understood from theoretical considerations. While for noninteracting particles, self-propulsion accelerates synchronization, the presence of excluded volume interactions gives rise to a richer scenario, where self-propulsion has a nonmonotonic impact on synchronization. We show that the synchronization of locally coupled mobile oscillators generically proceeds through coarsening, verifying the dynamic scaling hypothesis, with the same scaling laws as the 2D $XY$ model following a quench. Our results shed light into the generic nature of synchronization in time-dependent networks, providing an efficient way to understand more specific situations involving interacting mobile agents.

Popular Summary

Complex systems are formed by units whose individual evolution is simple but whose interactions give rise to a wide range of emergent collective behaviors. One of the paradigmatic examples of such emergent behavior is synchronization, in which a set of oscillators adjust their phases in response to the presence of other units, achieving a state where they all oscillate together. In nature, many systems that can be modeled as coupled oscillators—such as flocks of birds, groups of fireflies, or even robots—are not static but move around. Collections of individuals that possess the ability to propel themselves constitute a new class of physical systems known as “active matter.” The interplay between active motion and phase synchronization is crucial in the description of these types of systems. We develop a mathematical model that describes emergent cooperative behavior in such systems and brings together the effects of individual motion, their interactions, and layout of the network.

Our model considers the synchronization in time-evolving networks generated by the motion of self-propelled physical units (i.e., particles). We find that for noncolliding particles, self-propulsion accelerates synchronization. When collisions are considered, a much richer scenario appears for which an optimal self-propulsion speed exists. This approach lets us show that the synchronization generically proceeds through coarsening, where small, synchronized droplets gradually merge into bigger ones. The synchronization also exhibits the same dynamic scaling laws as the $XY$ model of magnetism, which describes ferromagnetism in two-dimensional materials.

While specific systems, such as flocks of birds or colonies of bacteria, have unique interactions, our model provides a general, efficient framework for understanding synchronization in self-propelling populations. Future work can use our model to understand specific situations and design synchronization strategies for autonomous robotics.

Figure 1

Evolution of the phase difference $C_{\theta }(t)/C_{\theta }(0)$ (after averaging over 500 independent runs) in lin-log scale for a system of $L=80$, $N=1000$ self-propelled point particles for $K=0.1$, $R_{\theta }=2$, and several values of $\tau$.

Figure 2

Synchronization time as a function of $\tau$ for several values of $R_{\theta }$ and $K=0.1$ for $N=1000$ and $L=80$. We also show $T_s$ in the mean-field limit by a horizontal line and the predicted value of saturation ${\tau }_{\mathrm{sat}}\approx 35.5$ by a vertical dotted line.

Figure 3

(a) Synchronization time as a function of $R_{\theta }$ for $K=0.1$ in a system of size $L=80$ with $N=1000$ particles. The horizontal points show the data obtained from our FS toy model (FSM). (b) $T_s$ as a function of $K$ for our toy model with $R_{\theta }=2$. The solid red line indicates the FS prediction, Eq. (8).

Figure 4

Finite-size behavior of $T_{s}K$ for several values of the coupling strength $K$ shown in the key with fixed $\tau =300$, $R_{\theta }=2$, and $\rho =0.15625$. The broken lines indicate the $T_s\sim N$ scaling expected in the SS regime. The FS toy model results are shown for comparison.

Figure 5

Configurations at various times during the evolution from an incoherent initial state of two systems with $K=0.001$ (top) and $K=0.1$ (bottom). The phase of the oscillators is represented as Schlieren patterns in which a color scale is associated to ${\sin }^2(2\theta )$. Each vortex emanates eight brushes of alternate black and white. In all cases, $L=320$, $N=16000$, $\tau =300$, and $R_{\theta }=2$. For visibility, the gray background refers to empty space.

Figure 6

Dynamic scaling for self-propelled point oscillators at $\rho =0.15625$ (top) and hard disks at $\varphi =0.12$ (bottom). (a) Space time correlation function $G(r,t)$ at different times $t=10$, 29, 100, 331, 1000, 3082 (from bottom to top) as a function of $r$ for $\tau =300$. The inset shows the scaling plot $G(r,t)$ against $r/\xi (t)$ together with the BPT scaling function. (b) Time evolution of $\xi (t)$ in log-log scale for several values of $\tau$. A linear growth is represented in dashed lines for comparison. The inset shows the rate of growth $\lambda$ as a function of $\tau$. (c) $G(r,t)$ at the same different times as (a) for a system of hard disks with $\tau =300$. Again, the inset shows the scaling plot $G(r,t)$ versus $r/\xi (t)$ confronted to the BPT scaling function. (d) Time dependence of the growing correlation length for several values of $\tau$. In all cases, $R_{\theta }=2$, $K=0.1$, $N=2000$, and $L=113.14$.

Figure 7

Snapshots of a system made of $N=16000$ self-propelled hard disks at two different times, $t=6019$ and 70 065, for $\tau =300$, $\varphi =0.12$, $L=320$ (top) and $\varphi =0.45$, $L=167$ (bottom). Each particle is represented by a disk whose color, as in Fig. 5, represents its phase by the amplitude of ${\sin }^2(2\theta )$.

Figure 8

Cluster size distribution for a system composed of $N=1000$ disks at $\varphi =0.45$ and several values of $\tau$. The percolation threshold in this system is at ${\tau }_c\approx 10$. The continuous line represents the distribution Eq. (16) with $m^{*}\approx 65$ and $\nu =1.7$. The broken line corresponds to the power-law behavior $P(m)\sim m^{-\nu }$ at percolation with $\nu =1.7$.

Figure 9

(a) Synchronization time $T_s$ times the number density $\rho$ as a function of $\tau$ for several values of $\varphi$ shown in the key. We present the results of $T_s$ normalized by $1/\rho$ since $1/\rho K$ introduces a time scale that can be absorbed in the time units. (b) Diffusion time for the same set of parameters as the ones in (a). The broken line represents the dilute limit behavior ${\sigma }^2/D\propto {\tau }^{-1}$. The data for $\varphi =0$ with $R_{\theta }=2$ and $\rho =0.15625$ are also shown as a reference.

Figure 10

Dependence of $\lambda$ with $\tau$. In the inset we show that $\lambda T_s\approx 35451$ with a reasonable degree of accuracy of 1%.