Normal form of $O(2)$ Hopf bifurcation in a model of a nonlinear optical system with diffraction and delay

In this paper we construct an $O(2)$-equivarint Hopf bifurcation normal form for a model of a nonlinear optical system with delay and diffraction in the feedback loop whose dynamics is governed by a system of coupled quasilinear diffusion equation and linear Schrödinger equation. The coefficients of the normal form are expressed explicitly in terms of the parameters of the model. This makes it possible to constructively analyze the phase portrait of the normal form and, based on the analysis, study the stability properties of the bifurcating rotating and standing waves.