Local dynamics of equations with large delay

The local dynamics of a differential equation with large delay is analyzed using the normal forms technique. It is shown that, in the critical cases, families of parabolic equations play the role of infinite-dimensional normal forms. It is demonstrated analytically that even a very simple first-order delay equation can have a complicated dynamical behavior. Methods for constructing classes of stable modes for such equations are described. The proposed methods are extended for the case of secondorder equations.