ИСТИНА

We consider parabolic functional-differential equation (FDE) with delay and a shift of spatial argument under periodic boundary condition \begin{equation*} \frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2} - u + K ( 1 + \gamma \cos u(x + \theta, t - T) ),\quad u(0, t) = u(2 \pi, t). \end{equation*} This problem arises while modeling of the nonlinear optical systems with nonlocal delayed feedback loop in case of thin ring aperture. Our aim is to study rotating waves branching off the homogeneous steady state as a result of Andronov-Hopf bifurcation. To this end, we propose to apply a transition to rotating coordinates, which allows us to reduce the dimension of the problem and obtain a stationary 1-D FDE. This FDE governs the shape of rotating waves and includes unknown temporal frequency as additional parameter. By utilizing implicit function theorem we have proven the existence and uniqueness of 1-D rotating waves in a circle under usual Andronov-Hopf bifurcation conditions and obtained the coefficients of the corresponding small parameter expansion. Stability conditions of these rotating waves are obtained by utilizing normal form theory for delayed parabolic FDE.