Athermal Dynamics of Strongly Coupled Stochastic Three-State Oscillators

We study the behavior of globally coupled ensembles of cyclic stochastic three-state units with transition rates from $i-1$ to $i$ proportional to the number of units in state $i$. Contrary to mean-field theory predictions, numerical simulations show significant stochastic oscillations for sufficiently large coupling strength. The order parameter characterizing units synchrony increases monotonically with coupling while the coherence of oscillations has a maximum at a certain coupling strength. We find the exact formulas for the stationary probability distribution and the order parameter.

Figure 1

Time series of the instantaneous occupation numbers oscillators $n_1$, $n_2$, $n_3$ in states 1, 2, 3, respectively, for $N=1000$, and different values of the coupling parameter: (a) $b=0$, (b) $b=0.1$ (Inset: Transition diagram in a single unit); (c) $b=1$, (d) $b=10$.

Figure 2

Gillespie simulations of coupled stochastic oscillators: coherence parameter $CP=\Delta \omega /\omega$ (a) and order parameter $R$ (b) vs. $b$ for $N={10}^2$, ${10}^3$, ${10}^4$. Solid line in (b) is (8). Inset: frequency $\omega$ and bandwidth $\Delta \omega$ vs $b$.

Figure 3

(a) State space of the three-oscillator system $N=3$; Vertices correspond to distinct states ($n_1$, $n_2$, $n_3$) of the system, and arrows indicate transitions among the states. Expressions at the arrows show the corresponding transition rates. (b)–(e) Stationary probability distributions for $N=14$ and $b=0.2$ (b), $b=0.8$ (c), $b=1$ (d), $b=2$ (e).