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

##### 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
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