Turbulence in the Ott-Antonsen equation for arrays of coupled phase oscillators (Q2794852)

The authors consider a one-dimensional array of non-locally coupled heterogeneous phase oscillators. The coupling term between oscillators \(i\) and \(j\) is \(\sin{(\theta_i-\theta_j+\alpha)}\) where \(\theta_i\) is the phase of oscillator \(i\). By taking the continuum limit and using the Ott/Antonsen ansatz they derive an integro-differential evolution equation for the spatially-dependent Kuramoto order parameter. They analytically find and determine the stability of the incoherent state and of partially coherent plane waves with different wave numbers. Increasing the coupling strength for \(\alpha<\pi/4\) they find the appearance of partially coherent plane waves in the Eckhaus scenario. Increasing the coupling strength for \(\alpha>\pi/4\) they find that the incoherent state becomes unstable but no partially coherent plane waves are stable, as they have become unstable via a Benjamin-Feir instability. In this region either phase or amplitude turbulence are numerically observed. A codimension-two point at \(\alpha=\pi/4\) organises the possible dynamics.