Quasiregular Concentric Waves in Heterogeneous Lattices of Coupled Oscillators

We study the pattern formation in a lattice of locally coupled phase oscillators with quenched disorder. In the synchronized regime quasiregular concentric waves can arise which are induced by the disorder of the system. Maximal regularity is found at the edge of the synchronization regime. The emergence of the concentric waves is related to the symmetry breaking of the interaction function. An explanation of the numerically observed phenomena is given in a one-dimensional chain of coupled phase oscillators. Scaling properties, describing the target patterns are obtained.

Figure 1

Simulation results in a two-dimensional lattice of $150\mathrm{x}150$ phase oscillators (1, 2) with nearest neighbor coupling, $ϵ=1$, periodic boundary conditions, homogeneous initial conditions, and (a) $\gamma =2$, $\sigma =0.029$, (b) $\gamma =2$, $\sigma =0.173$, (c) $\gamma =2$, $\sigma =0.433$, and (d) $\gamma =0.3$, $\sigma =0.433$. The random frequencies are taken from a uniform distribution of variance ${\sigma }^2$. Plotted is the sine of the phases ${\theta }_i$ as gray level. Similar results are obtained for open boundaries. (a)–(c): effect of increased heterogeneity; (c)–(d): influence of nonisochronicity $\gamma$. Insets show the natural frequencies ${\omega }_i$ as gray levels.

Figure 2

Phase differences ${\varphi }_i$ (left) and phase profile ${\theta }_i-{\theta }_1$ (b),(d) in units of $2\pi$ for a chain of 500 phase oscillators (4) with uniformly distributed frequencies ${\omega }_i\in [-0.1,0.1]$ and open boundaries. (a),(b) antisymmetric coupling (2) with $\gamma =0$. (c),(d) unidirectional coupling (5) with $\Gamma (\varphi )=\varphi \Theta (\varphi )$. (e) System (2) with $\gamma =2$. (f) Transfer map $T_{\Omega }({\varphi }_i)$ (8) for Eq. (2) with $\gamma =2$ (solid line) and fixed points ${\varphi }^{*}$ (solid circles).

Figure 3

Characterization of wave patterns by the locking frequency $\gamma \Omega$ (a), the wave length $\lambda /\gamma$ (b), the quality factor $Q_{\varphi }$ (c), and the cross correlation $r$ between neighboring phase differences (d). Numerical solutions are obtained by integrating system (1, 2) for one-dimensional (circles) and two-dimensional lattices (squares) in the synchronization regime. In one dimension the integrations were carried out with chains of length 500 and averaged over 50 simulation runs for $\gamma =2$ (red circles) and $\gamma =6$ (blue circles). In the two-dimensional system each point (green squares) represents one single simulation in a $100\mathrm{x}100$ lattice with $\gamma =2$. The results using the eigenvector method (12) are shown as (black $+$). Each point represents an average of 500 simulations with $N=256$. Further, indicated in (a),(b) are straight lines with a given exponent $\alpha$ and $\beta$ (dashed lines). The wavelength was obtained for one dimension as $\lambda =2\pi /{\varphi }^{*}$ and in the two-dimensional system from a Fourier analysis of the phase profile. The plateaus in the wavelength plot are finite size effects.