Dynamics of the Kuramoto equation with spatially distributed control

•Equation with spatially distributed control is studied by asymptotic methods.•Two distribution functions: almost symmetric and strongly asymmetric relative to zero.•For small control coefficient stability of running waves are studied.•For large control it is shown that dynamics described by behavior of quasi-normal form.

We consider the scalar complex equation with spatially distributed control. Its dynamical properties are studied by asymptotic methods when the control coefficient is either sufficiently large or sufficiently small and the function of distribution is either almost symmetric or significantly nonsymmetric relative to zero. In all cases we reduce original equation to quasinormal form – the family of special parabolic equations, which do not contain big and small parameters, which nonlocal dynamics determines the behaviour of solutions of the original equation.