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Twisted states in nonlocally coupled phase oscillators with bimodal frequency distribution. (English) Zbl 1514.34065

The authors study a ring of Kuramoto phase oscillators with nonlocal coupling governed by \[ \dot{\theta}_j=\omega_j+\frac{K}{2M+1}\sum_{k=-M}^M \sin{(\theta_{j+k}-\theta_j)} \] For intrinsic frequencies \(\{\omega_j\}\) chosen from a Lorentzian distribution, such a system is known to support partially coherent twisted waves for which the locally-averaged phase linearly varies with position around the ring. In this paper the intrinsic frequencies are randomly chosen from a superposition of two equal-height Lorentzian distributions. As well as the twisted waves mentioned above, the system now supports twisted standing waves, consisting of two travelling waves moving in opposite directions. The existence and stability of these twisted states is analysed exactly (in the limit of an infinite number of oscillators) using the Ott/Antonsen ansatz.

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D06 Synchronization of solutions to ordinary differential equations
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