Local dynamics of second-order differential equation with delayed derivative

We study the nonlinear dynamics of second-order differential equation with delayed feedback depending on the derivative. The problem in question contains a small multiplier at the highest derivative, so it is singularly perturbed. We determine the stability of equilibrium depending on the parameters and find critical (bifurcation) cases. In each critical case, asymptotic approximations for the spectrum points (roots of the characteristic equation) are determined. The main feature of the problem under consideration is that in critical cases the spectrum consists of two parts: an infinite chain of points that tend to the imaginary axis and one or two more points located near the imaginary axis.

Using methods of asymptotic analysis to study bifurcations, in the critical cases we construct special equations – quasinormal forms. Quasinormal form is an analog of normal form. It does not depends on small parameter and its solutions provide the main part of the asymptotic approximation of the solutions of the original problem. Each quasinormal form is a partial differential equation with an antiderivative operator and integral term in nonlinearity. For the constructed forms stable periodic solutions are determined, asymptotic approximations on stable periodic solutions of original problem is obtained and the bifurcations that occur are described.

Also, the situation where two successive bifurcations occur in the system was described.