Stochastic Kuramoto oscillators with discrete phase states

We present a generalization of the Kuramoto phase oscillator model in which phases advance in discrete phase increments through Poisson processes, rendering both intrinsic oscillations and coupling inherently stochastic. We study the effects of phase discretization on the synchronization and precision properties of the coupled system both analytically and numerically. Remarkably, many key observables such as the steady-state synchrony and the quality of oscillations show distinct extrema while converging to the classical Kuramoto model in the limit of a continuous phase. The phase-discretized model provides a general framework for coupled oscillations in a Markov chain setting.

Figure 1

(a) Schematic depiction of a phase-discretized oscillator with phase increment $\varepsilon =2\pi /m$ with $m\in \mathbb{N}$ and transition frequency $\omega$. (b) Stochastic trajectories of the phase $\varphi =\varepsilon \phi$ and the oscillatory signal $Rex=cos\varphi$ for a single oscillator for different values of $m$.

Figure 2

Synchronization transient for different values of $m$ for two coupled oscillators. (a) Kuramoto order parameter $\rho$, defined in Eq. (18); (b) imaginary part of the cross correlation $\chi \left(t\right)=\langle x_1\left(t\right)x_2^{*}\left(t\right)\rangle$. Dots show averages over stochastic trajectories of the phase-discretized model Eq. (8) with the coupling function Eq. (16); the red solid line shows the result for the classical Kuramoto model Eq. (1); and dashed horizontal lines show the exact steady-state solutions, given by Eqs. (24) and (C6), respectively. Parameters are ${\omega }_1=0.75$, ${\omega }_2=1.25$, $\kappa =1$, ${\varphi }_0=0$.

Figure 3

Synchronization time towards the steady state as a function of the phase discretization $m$ for two coupled oscillators. (a) Time ${\tau }_{\mathrm{h}}$ it takes to reach the order parameter $1/2$, defined by Eq. (20); (b) time ${\tau }_{\nu }$ it takes to reach a fraction $\nu$ of the steady-state order parameter $R$, defined by Eq. (21). System parameters as in Fig. 2.

Figure 4

Phase coherence of the two-oscillator system for different phase discretizations. Plots show the expectations values of the steady-state order parameter $R$ and the imaginary part of the cross correlation $X$ for different configurations. Solid lines show the exact solutions, given by Eqs. (24) and (C6), respectively; dots show time averages of simulated stochastic trajectories of length $T=2000$; dashed and dotted horizontal lines show the corresponding results for the classical Kuramoto model. (a, ${\mathrm{a}}^{\prime }$) Zero intrinsic frequencies: ${\omega }_1={\omega }_2=0$, $\kappa =1$, ${\varphi }_0=0$; (b, ${\mathrm{b}}^{\prime }$) Equal intrinsic frequencies: ${\omega }_1={\omega }_2=1$, ${\varphi }_0=0$, for $\kappa =1$ (dark) and $\kappa =0.25$ (light); (c, ${\mathrm{c}}^{\prime }$) Unequal frequencies: ${\omega }_1=0.75$, ${\omega }_2=1.25$, ${\varphi }_0=0$ with $\kappa =1$ (dark) and $\kappa =0.2$ (light); (d, ${\mathrm{d}}^{\prime }$) Nonzero phase shift: ${\omega }_1=0.75$, ${\omega }_2=1.25$, $\kappa =1$, for ${\varphi }_0=\pi$ (dark) and ${\varphi }_0=\pi /4$ (light). Insets show the corresponding exact solutions for a larger range of the phase discretization $m$ (note the logarithmic scale of the $m$ axis).

Figure 5

Steady-state order parameter $R$ as a function of the phase discretization $m$ for different coupling strengths $\kappa$ for two coupled oscillators, as given by Eq. (24). Dashed lines show the Kuramoto limit $m\to \infty$, given by Eq. (23). The other parameters are ${\omega }_1=0.75$, ${\omega }_2=1.25$, ${\varphi }_0=0$ as in Fig. 4.

Figure 6

Order parameter $R$ (left column) and mean quality $\mathcal{Q}$ (right column) as a function of the phase discretization $m$ and the coupling strength $\kappa$ for two coupled oscillators. The panels show the two cases of (a,b) equal frequencies, ${\omega }_1={\omega }_2=1$; and (c,d) unequal frequencies, ${\omega }_1=0.75$, ${\omega }_2=1.25$. The coupling phase shift is ${\varphi }_0=0$.

Figure 7

Synchronization and precision properties for systems of many oscillators. (a) Synchronization transient as indicated by the time-dependent Kuramoto order parameter $\rho$ [Eq. (18)] for different phase discretizations and numbers of oscillators. (b, c) Synchronization times as defined in Eqs. (20) and (21) as a function of the phase discretization for different numbers of oscillators, analogous to Fig. 3 for the case of two oscillators. (d, e) Steady-state order parameter $R$ and quality factor $Q$ as a function of the phase discretization for different numbers of oscillators for the full phase-discrete system (dots) and the linear noise approximation (curves), given by Eqs. (29, 30, 31). The $m$ axes in panels d and e are the same. System parameters are ${\omega }_i=1$, $\kappa =1$, ${\varphi }_0=0$.

Figure 8

Synchronization transition with the coupling strength as control parameter. The plot shows numerical results for the steady-state order parameter $R$ as a function of the coupling strength $\kappa$ for the model defined by Eqs. (27, 28, 29, 30, 31, 32) for different phase discretizations $m$ (colored dots) as well as for the linear noise approximation (LNA) given by Eqs. (29) and (30) (blue curve) and the deterministic Kuramoto model Eq. (1) (red curve). Simulations involve $N=500$ oscillators and averages are taken over 25 realizations of the frequency distribution.