Local solutions of the fast–slow model of an optoelectronic oscillator with delay

We study a difference–differential model of an optoelectronic oscillator that is a modification of the Ikeda equation with delay. We analyze the stability of the zero equilibrium state. We note that the number of roots of the characteristic equation of the linearized problem with the real part that is close to zero increases without bound for order parameter values tending to bifurcational ones. The asymptotics of such roots determines the asymptotic representation of solutions of the original problem that appear in a neighborhood of zero. An explicit change of variables allows finally obtaining equations of the special form for slow amplitudes that are independent of a small parameter and satisfy boundary conditions of the type of periodicity in one of the variables. It is convenient to consider such a variable as a spatial one, although the “fast” time plays this role. We determine amplitudes and frequencies of oscillatory components of the solutions. We formulate results about the correspondence between local solutions of the original system and nonlocal solutions of the partial differential equations playing the role of normal forms.