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Circular Cumulant Reductions for Macroscopic Dynamics of Kuramoto Ensemble with Multiplicative Intrinsic Noise

We demonstrate the application of the circular cumulant approach for thermodynamically large populations of phase elements, where the Ott-Antonsen properties are violated by a multiplicative intrinsic noise. The infinite cumulant equation chain is derived for the case of a sinusoidal sensitivity of the phase to noise. For inhomogeneous populations, a Lorentzian distribution of natural frequencies is adopted. Two-cumulant model reductions, which serve as a generalization of the Ott-Antonsen ansatz, are reported. The accuracy of these model reductions and the macroscopic collective dynamics of the system are explored for the case of a Kuramototype global coupling. The Ott-Antonsen ansatz and the Gaussian approximation are found to be not uniformly accurate for non-high frequencies.