Turbulence in the Ott-Antonsen equation for arrays of coupled phase oscillators. (English) Zbl 1338.34080

Wolfrum, M.; Gurevich, S. V.; Omel’chenko, O. E.

Nonlinearity 29, No. 2, 257-270 (2016).

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.

Reviewer: Carlo Laing (Auckland)

MSC:

34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D06	Synchronization of solutions to ordinary differential equations
35B36	Pattern formations in context of PDEs
45J05	Integro-ordinary differential equations

Keywords:

coupled oscillators; Ott/Antonsen; turbulence; Kuramoto model; twisted states; Benjamin-Feir instability; Eckhaus scenario