Spatially distributed control of the dynamics of the logistic delay equation

The influence exerted by a small spatially inhomogeneous control on the dynamics of the logistic delay equation is studied. This paper consists of two parts. The first deals with the case where the logistic delay equation has a stable relaxation cycle. It is shown that a small control function can give rise to complex relaxation objects, namely, to a large number of different attractors. In the second part, the local dynamics of the stability problem is analyzed in a neighborhood of equilibrium in a close-to-critical case of “infinite” dimension. Special quasi-normal forms are constructed whose nonlocal dynamics determine the local behavior of solutions to the original equation. Some results of a numerical analysis are presented.