Exact finite-dimensional reduction for a population of noisy oscillators

A description of complex systems by virtue of a few apposite order parameters is an indispensable tool of the theoretical analysis of nonequilibrium dynamics. In many cases, such a reduction is possible close to a bifurcation point, where a separation of timescales can be employed to derive approximate closed equations for a few order parameters. Here, we report on an exact reduction of the kinetic (generalized Fokker-Planck) equation to three complex variables. The dynamical equations are shown to contain the Ott-Antonsen dynamics as an attracting manifold. We demonstrate, how this finite-dimensional description allows for exact calculation of the effect of resetting of the ensemble.