Normalized boundary value problems in the model of optoelectronic oscillator delayed

Purpose of this work is reduction of differential-difference-model of optic-electronic oscillator to more simple normalized boundary value problems. We study the dynamics of an optoelectronic oscillator with delayed feedback in the vicinity of the zero equilibrium state. The differential-difference-model contains a small parameter with the derivative. It is shown that in a certain neighborhood of the bifurcation point, the number of roots of the characteristic equation that have a real part close to zero increases unlimitedly with decreasing small parameter. Partial boundary value problems are obtained that play the role of normal forms for the original system and which have stationary solutions in the form of symmetric or asymmetric rectangular structures. The multistability of rectangular structures with a different number and shape of steps is shown. The spatio-temporal representation of solutions of the initial equation with delay is substantiated. The frequencies and amplitudes of oscillating solutions of the delay equation are determined. Research methods. We apply standard methods of normal forms on central manifolds, as well as special methods for infinite-dimensional normalization. An algorithm is proposed for reducing the initial delayed equation to the boundary-value problem for slowly varying amplitudes. Results. Finite-dimensional and special infinite-dimensional equations - boundary value problem are constructed that play the role of normal forms. Their nonlocal dynamics determines the behavior of solutions to the original equation with delay from a small neighborhood of the equilibrium. Asymptotic formulas for solutions on the interval $[t_{0},\infty)$ are given.