Dynamics of mobile coupled phase oscillators

We study the transient synchronization dynamics of locally coupled phase oscillators moving on a one-dimensional lattice. Analysis of spatial phase correlation shows that mobility speeds up relaxation of spatial modes and leads to faster synchronization. We show that when mobility becomes sufficiently high, it does not allow spatial modes to form and the population of oscillators behaves like a mean-field system. Estimating the relaxation timescale of the longest spatial mode and comparing it with systems with long-range coupling, we reveal how mobility effectively extends the interaction range.

Figure 1

Mobile oscillators on a one-dimensional lattice. (a) Bars indicate a coupling range $r=2$ for oscillators in the bulk and at a boundary. (b) During a small time interval $\Delta t$, each oscillator exchanges its position with one of its adjacent neighbors with probability $\lambda \Delta t$.

Figure 2

Snapshots of spatial phase profiles. (a) $\lambda /\kappa =0$, (b) $\lambda /\kappa =10$, and (c) $\lambda /\kappa ={10}^4$. Each dot represents an oscillator. In all panels, $\kappa =1$, $r=1$, and $N=100$.

Figure 3

Time evolution of the correlations $\rho$ defined by Eq. (2), (a)–(c), and $1-\rho$, (d)–(f). $\lambda /\kappa =0$ in (a) and (d), $\lambda /\kappa =10$ in (b) and (e), $\lambda /\kappa ={10}^4$ in (c) and (f). Lines of different colors represent correlations for different distances $d$, as indicated in the figure. In all panels, $\kappa =1$, $r=1$, and $N=100$; from left to right, the lines are $\rho \left(1\right)$, $\rho \left(5\right)$, $\rho \left(20\right)$, $\rho \left(40\right)$, and $\rho \left(99\right)$.

Figure 4

Onset of nonlocal and mean-field behavior. (a) Dependence of the characteristic time $T_c$ on $\lambda /\kappa$ for different system sizes. (b) Dependence of the scaled characteristic time ${\tilde{T}}_c$ on $\lambda /\kappa$. The solid line indicates Eq. (3). The dashed line indicates ${\tilde{T}}_c=1$. (c) Dependence of $T_c$ on the scaled moving rate $(\lambda /\kappa )/N^2$. In (a) and (c), the dashed line indicates $T_c=1/2$. In all panels, $\kappa =1$ and $r=1$.

Figure 5

Effect of a single exchange event on the longest spatial mode. (a) Shape of the longest spatial mode and a small defect caused by an exchange event between two neighboring oscillators at time $t=t_0+{\tau }_1$. Each dot indicates an oscillator. See text for symbols. (b) Dependence of $1-{\langle c_{11}\rangle }_x$ on system size $N$ for different coupling ranges $r$. Symbols indicate the results obtained from a numerical calculation where we first compute $c_{11}\left(x\right)$ for each $x$ and then compute its spatial average ${\langle c_{11}\rangle }_x$. The broken line is ${\pi }^2/N^3$, as in Eq. (17).

Figure 6

Effective coupling range induced by mobility. (a) Dependence of the characteristic time $T_c$ on the coupling range $r$ for nonmobile oscillators in systems of different sizes $N$. Symbols correspond to numerical simulations, lines indicate the approximation given by Eq. (26). Parameters: $\kappa =1$ and $\lambda /\kappa =0$. (b) Dependence of the effective coupling range $r_e$ given by Eq. (27) on $\lambda /\kappa$. $r=1$. The dotted line represents $r_e=\sqrt{48}/4\times \sqrt{\lambda /\kappa }$.

Figure 7

Onset of nonlocal behavior for mobile oscillators with coupling ranges $r$ and their effective coupling range $r_e$. (a) Dependence of $T_c$ on $\lambda /\kappa$ for different coupling ranges $r$. Symbols indicate $T_c$ obtained by numerical simulations of Eq. (1), while each colored line indicates Eq. (23) for corresponding $r$ of the same color. The horizontal dotted line indicates $T_c=1/2$. $\kappa =1$ and $N=100$. (b) Dependence of the effective coupling range $r_e$ given by Eq. (31) on $\lambda /\kappa$. The dotted line represents $r_e=\sqrt{48}/4\times \sqrt{\lambda /\kappa }$. From top to bottom, the solid lines are $r=20,10,5,2,1$.