Features of the local dynamics of the opto-electronic oscillator model with delay

E. V. Grigoryeva ; S. A. Kashchenko ; D. V. Glazkov

Modelirovanie i analiz informacionnyh sistem, Tome 25 (2018) no. 1, pp. 71-82

Résumé

We consider electro-optic oscillator model which is described by a system of the delay differential equations (DDE). The essential feature of this model is a small parameter in front of a derivative that allows us to draw a conclusion about the action of processes with different order velocities. We analyse the local dynamics of a singularly perturbed system in the vicinity of the zero steady state. The characteristic equation of the linearized problem has an asymptotically large number of roots with close to zero real parts while the parameters are close to critical values. To study the existent bifurcations in the system, we use the method of the behaviour constructing special normalized equations for slow amplitudes which describe of close to zero original problem solutions. The important feature of these equations is the fact that they do not depend on the small parameter. The root structure of characteristic equation and the supercriticality order define the kind of the normal form which can be represented as a partial differential equation (PDE). The role of the ”space” variable is performed by ”fast” time which satisfies periodicity conditions. We note fast response of dynamic features of normalized equations to small parameter fluctuation that is the sign of a possible unlimited process of direct and inverse bifurcations. Also, some obtained equations possess the multistability feature.

Détail

Keywords: differential equation, local dynamics, small parameter, asymptotics, normal form, boundary value problem.
Mots-clés : bifurcation