Local Dynamics of a Model of an Opto-Electronic Oscillator with Delay

Resumo

We consider the dynamics of an electro-optic oscillator described by a system of differential equations with delay. An essential feature of this model is that one of the derivatives is multiplied by a small parameter, which allows us to draw conclusions about the action of processes with rates of different orders. The local dynamics of a singularly perturbed system near a zero steady state is analyzed. When values of the parameters are close to critical, the characteristic equation of the linearized problem has an asymptotically large number of roots with close-to-zero real parts. The bifurcations taking place in the system are studied by constructing special normalized equations for slow amplitudes, which describe the behavior of close-to-zero solutions of the initial problem. An important feature of these equations is that they do not depend on the small parameter. The structure of roots of the characteristic equation and the order of supercriticality determine the normal form, which can be represented by a partial differential equation. The “fast” time, for which periodicity conditions are fulfilled, acts as a “spatial” variable. The high sensitivity of dynamic properties of normalized equations to the change of a small parameter is noted, which is a sign of a possible unlimited process of direct and inverse bifurcations. Also, the obtained equations exhibit multistability of solutions.