Hole structures in nonlocally coupled noisy phase oscillators

We demonstrate that a system of nonlocally coupled noisy phase oscillators can collectively exhibit a hole structure, which manifests itself in the spatial phase distribution of the oscillators. The phase model is described by a nonlinear Fokker-Planck equation, which can be reduced to the complex Ginzburg-Landau equation near the Hopf bifurcation point of the uniform solution. By numerical simulations, we show that the hole structure clearly appears in the space-dependent order parameter, which corresponds to the Nozaki-Bekki hole solution of the complex Ginzburg-Landau equation.

Figure 1

Dependence of the order-parameter modulus on the noise strength, obtained from the Langevin-type equation (1) using different values of the number of oscillators $N$. The data are plotted using rescaled variables, revealing the scaling function given by Eq. (13).

Figure 2

Instantaneous spatial profile of the phase distribution function $f(\psi ,x,t)$ obtained from a numerical simulation of the nonlinear Fokker-Planck equation (4).

Figure 3

(Color online) Instantaneous spatial profile of the order-parameter modulus (a). Instantaneous phase portrait of the order parameter (b). They are obtained from Fig. 2 using the relation given in Eq. (12). Solid lines are drawn using the Nozaki-Bekki hole solution of the CGLE given by Eqs. (22, 23), where the parameter values are estimated as $a_{\mathrm{FP}}\simeq 0.45$, $b_{\mathrm{FP}}\simeq 0.23$, $c_{\mathrm{FP}}\simeq 26$, and $s_{\mathrm{FP}}\simeq 0.15$.

Figure 4

Instantaneous spatial profile of the local oscillator phase obtained from a numerical simulation of the Langevin-type equation (1).