Collective dynamics of coupled oscillators with random pinning

Abstract We analyze a large system of nonlinear oscillators with random pinning, mean-field coupling and external drive. For small coupling and drive strength, the system evolves to an incoherent pinned state, with all the oscillators stuck at random phases. As the coupling or drive strength is increased beyond a depinning treshold, the steady-state solution switches to a coherent moving state, with all the oscillators moving nearly in phase. This depinning transition is discontinuous and hysteretic. We also show analytically that there is a delayed onset of coherence in response to a sudden superthreshold drive. The time delay increases as the threshold is approached from above. The discontinuous, hysteretic transition and the delayed onset of coherence are directly attributable to the form of the coupling, which is periodic in the phase difference between oscillators. The system studied here provides a simple model of charge-density wave transport in certain quasi-one-dimensional metals and semiconductors in the regime where phase-slip is important; however this paper is intended primarily as a study of a model system with analytically tractable collective dynamics.

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