Turbulence in the Ott–Antonsen equation for arrays of coupled phase oscillators

In this paper we study the transition to synchrony in an one-dimensional array of oscillators with non-local coupling. For its description in the continuum limit of a large number of phase oscillators, we use a corresponding Ott–Antonsen equation, which is an integro-differential equation for the evolution of the macroscopic profiles of the local mean field. Recently, it was reported that in the spatially extended case at the synchronisation threshold there appear partially coherent plane waves with different wave numbers, which are organised in the well-known Eckhaus scenario. In this paper, we show that for Kuramoto–Sakaguchi phase oscillators the phase lag parameter in the interaction function can induce a Benjamin–Feir-type instability of the partially coherent plane waves. The emerging collective macroscopic chaos appears as an intermediate stage between complete incoherence and stable partially coherent plane waves. We give an analytic treatment of the Benjamin–Feir instability and its onset in a codimension-two bifurcation in the Ott–Antonsen equation as well as a numerical study of the transition from phase turbulence to amplitude turbulence inside the Benjamin–Feir unstable region.