We consider a system of globally coupled phase-only oscillators with distributed intrinsic frequencies and evolving in the presence of distributed Gaussian white noise, namely, a Gaussian white noise whose strength for every oscillator is a specified function of its intrinsic frequency. In the absence of noise, the model reduces to the celebrated Kuramoto model of spontaneous synchronization. For two specific forms of the mentioned functional dependence and for a symmetric and unimodal distribution of the intrinsic frequencies, we unveil the rich long-time behavior that the system exhibits, which stands in stark contrast to the case in which the noise strength is the same for all the oscillators, namely, in the studied dynamics, the system may exist in either a synchronized, or an incoherent, or a time-periodic state; interestingly, all these states also appear as long-time solutions of the Kuramoto dynamics for the case of bimodal frequency distributions, but in the absence of any noise in the dynamics.
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