Perturbation analysis of the Kuramoto phase-diffusion equation subject to quenched frequency disorder

The Kuramoto phase-diffusion equation is a nonlinear partial differential equation which describes the spatiotemporal evolution of a phase variable in an oscillatory reaction-diffusion system. Synchronization manifests itself in a stationary phase gradient where all phases throughout a system evolve with the same velocity, the synchronization frequency. The formation of concentric waves can be explained by local impurities of higher frequency which can entrain their surroundings. Concentric waves in synchronization also occur in heterogeneous systems, where the local frequencies are distributed randomly. We present a perturbation analysis of the synchronization frequency where the perturbation is given by the heterogeneity of natural frequencies in the system. The nonlinearity in the form of dispersion leads to an overall acceleration of the oscillation for which the expected value can be calculated from the second-order perturbation terms. We apply the theory to simple topologies, like a line or sphere, and deduce the dependence of the synchronization frequency on the size and the dimension of the oscillatory medium. We show that our theory can be extended to include rotating waves in a medium with periodic boundary conditions. By changing a system parameter, the synchronized state may become quasidegenerate. We demonstrate how perturbation theory fails at such a critical point.

Figure 1

(Color online) Influence of frequency disorder in a one-dimensional lattice. Plotted is the shift of the synchronization frequency $\Omega$ from the frequency of the synchronized state with identical oscillators, ${\Omega }^0$, as a function of the oscillator heterogeneity $\sigma$. We compare simulations of the discrete Kuramoto model, Eq. (5), for a chain of $N$ oscillators with the second-order perturbation theory, Eq. (37), with $L\to N$. Each data point from the simulations is an average value from 50 runs with different realizations of iid. random frequencies $(\mathbb{E}\left[{\eta }_k^2\right]=1)$, where for each realization the average frequency has been shifted to zero ${⟨\eta ⟩}_{\text{System}}=0$. (a) Comparison of the results for a ring of $N=100$ nonisochronous $(\gamma =0.25)$ phase oscillators, without topological charge (blue line and plus marks) and with a topological charge of $l=10$ (red line and × marks). (b) Comparison of rings of different sizes $N=16$ (red line and × marks) and $N=100$ (blue line and plus marks) oscillators, but with the same nonisochronicity $\gamma =0.25$ and topological charge $l=3$.

Figure 2

Quasiregular wave patterns (a) in a rectangular medium with periodic boundary conditions and (b) on the surface of a sphere. We have used the discrete Kuramoto model, Eq. (5), on (a) a grid of $150\times 150$ phase oscillators and (b) on an almost homogeneous discretization of the sphere surface with 20 480 points on the faces of a triangular tessellation [25]. The natural frequencies of the individual oscillators were independently uniformly distributed with standard deviation $\sigma =0.2$. The topology of the square lattice in (a) is that of a 2-torus, and it is possible to have topological charges without phase singularities (large phase differences). We used an initial condition with topological charges $l_x=3$ and $l_y=7$. Shown is the sine of the phases in gray levels after a transient time to synchronization.

Figure 3

(Color online) Synchronization analysis in a one-dimensional chain of $N=200$ nonidentical Kuramoto phase oscillators with two competing pacemakers regions according to Eq. (48). (b) Frequency values $-{\eta }_n$ corresponding to the potential of the discrete Hamiltonian (values ${\eta }_n$ are shifted so that the mean ${⟨\eta ⟩}_{\text{System}}=0$ is exactly zero). There are two potential wells at which the ground state can be localized, a deeper well on the left and a shallower but broader well on the right. (a) and (c) Largest eigenvalues (solid blue lines) of the negative Hamiltonian $-\mathbf{H}$ in dependence on the heterogeneity $\sigma$, the second-order perturbation approximation [Eq. (29), dashed red line], and the value ${\sigma }_{\mathit{cr}}$ for which the ground state becomes quasidegenerate (dashed black line). The quality of the approximation can be seen in double-logarithmic scales in (c). (d)–(f) Numerically determined ground-state eigenvectors in a semilogarithmic scale for (d) $\sigma =0.03$, (e) $\sigma ={\sigma }_{\mathit{cr}}=0.036917$ near the point of quasidegeneracy, and (f) $\sigma =0.04$. Exponential localization at a potential well corresponds to concentric waves around this pacemaker region in the Kuramoto phase- diffusion equations [15, 16].

Figure 4

(Color online) Synchronization analysis in a one-dimensional chain of $N=200$ nonidentical Kuramoto phase oscillators according to Eq. (48) and independent, identically, uniformly distributed random frequencies ${\eta }_n$ (values are shifted so that the mean ${⟨\eta ⟩}_{\text{System}}=0$ is exactly zero). (b) Frequency values $-{\eta }_n$ corresponding to the potential of the discrete Hamiltonian (solid blue line) and a Gaussian filtering of width 2 (bold red line). There are several potential regions at which the ground state could be localized. (a) and (c) Largest eigenvalues (solid blue lines) of the negative Hamiltonian $-\mathbf{H}$ in dependence on the heterogeneity $\sigma$, the second-order perturbation approximation [Eq. (29), dashed red line], and the value ${\sigma }_{\mathit{cr}}$ for which the ground state becomes quasidegenerate (dashed black line). The quality of the approximation can be seen in double-logarithmic scales in (c). (d)–(f) Numerically determined ground-state eigenvectors in a semilogarithmic scale for (d) $\sigma =0.04$, (e) $\sigma ={\sigma }_{\mathit{cr}}=0.065135$ near the point of quasidegeneracy, and (f) $\sigma =0.07$. Exponential localization at a potential well corresponds to concentric waves around this pacemaker region in the Kuramoto phase-diffusion equations [15, 16].

Figure 5

(Color online) Synchronization analysis in a one-dimensional chain of $N=200$ nonidentical Kuramoto phase oscillators with periodic boundary conditions and independent, identically distributed normal random frequencies ${\eta }_n$ (values are shifted so that the mean ${⟨\eta ⟩}_{\text{System}}=0$ is exactly zero). The topological charge is $l=4$. (a) Three largest eigenvalue real parts $\mathrm{Re}{E}_0>\mathrm{Re}{E}_1>\mathrm{Re}{E}_1$ of the difference operator in Eq. (48) under variation of the heterogeneity $\sigma$. The second and third eigenvalues in this example are complex conjugated for $\sigma <0.09$, and the ground state becomes quasidegenerate for ${\sigma }_{\mathit{cr}}\approx 0.154$. (b) Unperturbed $(\sigma =0)$ rotating wave solution on a ring of oscillators, here as the sine of the phase (in gray levels) on the side of a cylinder for illustration. (c), (d) Stationary phase profiles of the corresponding discrete KPE [Eq. (5), blue lines] for $\sigma =0.15$ and $\sigma =0.16$, respectively, and the dashed black line is the unperturbed constant phase gradient.