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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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