Fronts and bumps in spatially extended Kuramoto networks
The author considers two models of coupled phase oscillators, using the continuum limit description and the Ott-Antonsen approach. The first model is a chain of \(N\) phase oscillators equally spaced on the one-dimensional spatial domain \([0,L]\) \[ \frac{d \theta_j}{d t} = \omega_j + k \operatorname{Im}( z_j(t) e^{-i \theta_j} ),\quad j=1,\dotsc,N, \tag{1} \] and coupled via a complex, non-local mean-field \(z_m\) given by \[ \begin{aligned}\frac{d z_m(t)}{d t} = - z_m(t) + \frac{L}{N} \sum\limits_{j=1}^N K_{mj} e^{i \theta_j(t)}, \\ K_{mj} = K\left( \frac{L |m - j|}{N - 1} \right),\quad K(x) = (1/2) e^{- |x|}.\end{aligned} \tag{2} \] The natural frequencies \(\omega_j\) of the oscillators are chosen from a Lorentzian distribution \(g(\omega) = \pi^{-1} [ (\omega - \omega_0)^2 + 1 ]^{-1}\) centred at \(\omega = \omega_0\), and \(k>0\) is an overall coupling strength. In the continuum limit \(N\to\infty\), the dynamics of system (1), (2) can be effectively described in terms of the probability density function \(F(x,\omega,\theta,t)\) (the probability that an oscillator at position \(x\) has intrinsic frequency \(\omega\) and state \(\theta\), at time \(t\)) that obeys some evolution integro-differential equation. It is shown that, for certain parameter values, the latter equation exhibits bistability between the completely asynchronous state (with constant probability distribution) and spatially uniform partially synchronous state (with an explicitly known, time-periodic probability distribution). Based on this observation, the author constructs moving front solutions, which connect regions of high local synchrony with regions of complete asynchrony, and bump solutions consisting of spatially-localized regions of partially-synchronous oscillators surrounded by complete asynchrony. Numerical continuation is used to determine parameter regions in which such solutions exist and are stable. An extension of the results to two spatial dimensions is discussed. The second model considered in the paper reads \[ \frac{d \theta_j}{d t} = \omega_j + \sigma_j \operatorname{Im}( R_j(t) e^{-i \theta_j} ),\quad j=1,\dotsc,N, \tag{3} \] where \[ R_m(t) = \frac{L}{N} \sum\limits_{j=1}^N K_{mj} e^{i \theta_j(t)}, \tag{4} \] and \(K_{mj}\) and \(\omega_j\) are defined as above. Instead of constant coupling strength present in the first model, here \(\sigma_j\) are chosen from the double-\(\delta\) distribution \[ \Gamma(\sigma) = (1 - p) \delta(\sigma - \sigma_1) + p \delta(\sigma - \sigma_2), \] where \(\sigma_1<0\) and \(\sigma_2>0\), and \(0\leq p\leq 1\) is a probability. For some parameter values, system (3), (4) was found to exhibit bistability. This fact is used to show that system (3), (4) supports moving fronts with positive, negative, or zero velocity, although without bumps.
scientific article; zbMATH DE number 5994489