We analyze a large system of nonlinear phase oscillators with sinusoidal nonlinearity, uniformly distributed natural frequencies and global all-to-all coupling, which is an extension of Kuramoto's model to second-order systems. For small coupling, the system evolves to an incoherent state with the phases of all the oscillators distributed uniformly. As the coupling is increased, the system exhibits a discontinuous transition to the coherently synchronized state at a pinning threshold of the coupling strength, or to a partially synchronized oscillation coherent state at a certain threshold below the pinning threshold. If the coupling is decreased from a strong coupling with all the oscillators synchronized coherently, this coherence can persist until the depinning threshold which is less than the pinning threshold, resulting in hysteretic synchrony depending on the initial configuration of the oscillators. We obtain analytically both the pinning and depinning threshold and also expalin the discontinuous transition at the thresholds for the underdamped case in the large system size limit. Numerical exploration shows the oscillatory partially coherent state bifurcates at the depinning threshold and also suggests that this state persists independent of the system size. The system studied here provides a simple model for collective behaviour in damped driven high-dimensional Hamiltonian systems which can explain the synchronous firing of certain fireflies or neural oscillators with frequency adaptation and may also be applicable to interconnected power systems.
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