Reduced cumulant models for macroscopic dynamics of Kuramoto ensemble with multiplicative intrinsic noise

The purpose of this work is developing reduced models describing the macroscopic dynamics of the Kuramoto ensemble with multiplicative intrinsic noise on the basis of the method of circular cumulants. Methods. The dynamics of the system is considered within the framework of the phase reduction. The dynamics equations are obtained by the method of circular cumulants. Stability of the asynchronous state is considered on the basis of linear analysis. Results are verified by the numerical simulation. Results. The infinite cumulant equation chain is derived for the Kuramoto ensemble with intrinsic multiplicative noise. Two closures of the cumulant series are proposed to construct reduced models of the ensemble dynamics. Conclusion. For a phase oscillator population with Kuramoto global coupling, the case of a multiplicative noise converges to the case of an additive one only in the high-frequency limit. Moreover, for low frequencies, the instability of the asynchronous state to formation of a macroscopic collective mode becomes monotonous. Two-cumulant model reductions provide a reasonable accuracy for the macroscopic description of the population dynamics. Meanwhile, the Ott-Antonsen ansatz and the Gaussian approximation fail to represent the system dynamics accurately for non-high frequencies.