Bifurcations of solutions to equations with deviating spatial arguments

A periodic boundary-value problem for an equation with deviating spatial argument is considered. This equation describes the phase of a light wave in light resonators with distributed feedback. Optical systems of this type are used in computer technologies and in the study of laser beams. The boundary-value problem was considered for two values of spatial deviations. In the work, bifurcation problems of codimensions $1$ and $2$ were analyzed by various methods of studying dynamical systems, for example, the method of normal Poincaré–Dulac forms, the method of integral manifolds, and asymptotic formulas. The problem on the stability of certain homogeneous equilibrium states is examined. Asymptotic formulas for spatially inhomogeneous solutions and conditions for their stability are obtained.