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Synchronization transition of identical phase oscillators in a directed small-world network
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FIG. 1.

The network model. Unidirectional small-world networks with nodes at [(a) and (b)] low shortcut density and [(c) and (d)] higher shortcut density . Joints of the network, i.e., nodes that receive more than one input, are marked gray. (b) At low shortcut densities most joints couple indirectly to two other joints through linear chain segments of length . (d) At high shortcut densities each joint couples to neighbors which are also with high probability joints.

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FIG. 2.

Synchronization transition in the parameter plane. Each point corresponds to an ensemble average over ten network realizations and time average over 600 units after an initial transient of 200. Shown are (a) the mean order parameter , (c) the mean oscillator frequency, and (d) the variance of phase velocities in the case of normalized input strength. For comparison we also show (b) the mean order parameter for non-normalized coupling strength for which a larger area of partial synchronization is observed at intermediate shortcut densities.

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FIG. 3.

Variance and distribution of phase velocities in the incoherent state. (a) Variance of phase velocities obtained from simulations with and (crosses) and from the simulations presented in Fig. 2(d) (circles) at . (b) The distribution of phase velocities in the incoherent state at (dots) is centered around the mean of . It is peaked at the center and much broader than a Gaussian distribution of the same variance (dashed line).

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FIG. 4.

Snapshots of the first 200 phases in a system of oscillators in a dynamical (quasi) equilibrium state for low shortcut density in (a) and (b) and high shortcut density in (c) and (d). (a) and (c) show the stable incoherent state in a parameter region of bistability with the partially synchronized states shown in (b) and (d). Only at low shortcut densities the phases have a spatiotemporal structure at the length scale of the chain segments.

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FIG. 5.

Bifurcation diagram of the order parameter as a function of control parameter at selected values of shortcut densities . Points on the branches of unstable (open squares) and stable (crosses) partially synchronized states were obtained as averages of the trajectory [see light gray area in (d)] under the control scheme given by Eq. (6). The green lines are sixth order polynomial fits constrained to because of the assumption of a Hopf bifurcation of the incoherent state. From these fits we also find the threshold for complete synchronization and the points of saddle node bifurcations of stable and unstable partially synchronized states. The dots in (a) are the average order parameter in simulation with networks of oscillators. Each point is an ensemble average of the order parameter over 50 realizations after 1000 units of time.

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FIG. 6.

Numerically determined synchronization points for (a) low shortcut densities, [(b) and (c)] intermediate shortcut densities, and (d) large shortcut densities. The open circles and the upward and the downward triangles mark the Hopf bifurcation points of the incoherent state, the transition points to complete synchronization, and the saddle-node bifurcation points , respectively. Stable partial synchronization is found between and . At intermediate to large shortcut densities [(c) and (d)] the transition to synchronization is very well-described by (solid line). This is not the case for very low shortcut densities (a) where approaches zero more slowly than linearly. The line of slope 0.5 in the double-logarithmic plot (a) is drawn for comparison. The critical line obtained from the heuristic mean field ansatz equation (13) [dashed line in (d)] agrees qualitatively with the asymptotic approach of to but is larger than the values obtained by our control scheme (open circles). The color code for the background of (b) and (c) is the same as in Figs. 2(a) and 2(b).

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FIG. 7.

Finite size scaling analysis of the nonequilibrium transition from partial to complete synchronization at for . In (a) simulation runs with system sizes up to were performed. At the critical point complete synchronization merges with the metastable partially synchronized state which is approached as if the transition is of mean field directed percolation universality. The line is drawn for comparison. The inset shows the linear approach of the mean order parameter in the vicinity of the critical point. The line of slope 1.01 is a linear fit to the data in double-logarithmic scales. Other critical exponents are obtained from the time statistics for a realization to reach the absorbing state of complete synchronization. (b) shows the median of this time at the critical point for various system sizes. In (c) we plot the fraction of 100 realizations which reach complete synchronization before the time for different as a function of . This defines a median , shown in (d), which approaches the critical point at a power law with exponent of −0.55 as a function of .

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FIG. 8.

Complex correlation function in the incoherent state. (a) Absolute value in logarithmic scales as a function of the distance on the ring backbone of the network for (blue crosses), (red circles), and (green squares). (b) Angle as a function of the distance on the ring backbone of the network for (blue crosses), (red circles), and (green squares). We used , well above the synchronization threshold. The dashed lines in (a) and (b) mark the mean distance in the network of size . (c) Scaling of absolute value with the number of neighbors at distance (blue crosses), (red triangles), and (green diamonds). (d) Logarithm of autocorrelation function at time difference for (blue crosses), (red circles), and (green triangles) and parametric fit to autocorrelation function of Brownian flight on the circle [Eq. (11), dashed lines]. See Table I for values of and . The inset shows the collapse of the curves under a rescaling of time with factor .

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FIG. 9.

Effective phase diffusion constant of a phase oscillator with complex state variable coupled to a complex valued Ornstein–Uhlenbeck process of unit variance and phase diffusion constant [Eq. (A8)]. The fixed point at is expected to be close to the rescaled effective phase diffusion constant of the characteristic stationary phase diffusion process in the incoherent state for .


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Table I.

Time scales of the chaotic phase diffusion process for various large mean degrees . We find that the effective phase diffusion constant and the effective scattering rate scale with and the variance of the phase velocities scales as [see Fig. 3(b)]. The transition point to synchronization has been determined with the help of our control scheme.


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We numerically study a directed small-world network consisting of attractively coupled, identical phase oscillators. While complete synchronization is always stable, it is not always reachable from random initial conditions. Depending on the shortcut density and on the asymmetry of the phase coupling function, there exists a regime of persistent chaotic dynamics. By increasing the density of shortcuts or decreasing the asymmetry of the phase coupling function, we observe a discontinuous transition in the ability of the system to synchronize. Using a control technique, we identify the bifurcation scenario of the order parameter. We also discuss the relation between dynamics and topology and remark on the similarity of the synchronization transition to directed percolation.


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Scitation: Synchronization transition of identical phase oscillators in a directed small-world network