System size synchronization

In this paper we bring out the existence of a kind of synchronization associated with the size of a complex system. A dichotomic random jump process associated with the dynamics of an externally driven stochastic system with $N$ coupled units is constructed. We define an output frequency and phase diffusion coefficient. System size synchronization occurs when the average output frequency is locked to the external one and the average phase diffusion coefficient shows a very deep minimum for a range of system sizes. Analytical and numerical procedures are introduced to study the phenomenon, and the results describe successfully the existence of system size synchronization.

Figure 1

Stochastic trajectories of the global variable $X\left(t\right)$ for $N=50$ (a), $N=250$ (b), and $N=900$ (c). The remaining parameter values are $\epsilon =2.7, D=0.7, A=0.03$, and $\omega =2\pi /T=0.001$. (d) shows the external driving force conveniently amplified.

Figure 2

Sketch of the filtering process for a system with size $N=250$ and the same parameter values as in Fig. 1. (a) depicts a random trajectory and the corresponding threshold values $X_{\mathrm{th}}$ and $-X_{\mathrm{th}}$. The filtered trajectory $\chi \left(t\right)$ is depicted in (b). (c) shows the external driving force conveniently amplified.

Figure 3

Plot of the ratio of the averaged output frequency, ${\mathrm{\Omega }}_{\mathrm{out}}$, and the driving one, $\omega$, vs the system size $N$ for $A=0.03, D=0.7, \omega =0.001$, and $\epsilon =2.7$. Filled circles indicate the results obtained from the simulation of the $N$ coupled Langevin equations in Eq. (1). Stars correspond to the results obtained from the simulation of the effective Langevin equation in Eq. (6). The solid line describes the analytical result obtained from Eq. (4). For the above parameter values, the rates of escape in Sec. 3 are given approximately by ${\gamma }_{+}=0.04e^{-0.08N}$ and ${\gamma }_{-}=0.03e^{-0.01N}$.

Figure 4

Plot of the logarithm of the averaged phase diffusion coefficient, ${\log }_{10}\left(D_{\mathrm{out}}\right)$, vs the system size $N$ for $A=0.03, D=0.7, \omega =0.001$, and $\epsilon =2.7$. Filled circles indicate the results obtained from the simulation of the $N$ coupled Langevin equations in Eq. (1). Stars correspond to the results obtained from the simulation of the effective Langevin equation in Eq. (6). The solid line describes the analytical result obtained from Eq. (5). For the above parameter values, the rates of escape in Sec. 3 are given approximately by ${\gamma }_{+}=0.04e^{-0.08N}$ and ${\gamma }_{-}=0.03e^{-0.01N}$.