Quasinormal forms for systems of two equations with large delay

A system of two equations with delay is considered. The purpose of this work is to study the local dynamics of this system under the assumption that the delay parameter is sufficiently large. Critical cases in the problem of stability of an equilibrium state are identified and it is shown that they have infinite dimension. Methods. The research is based on the use of special methods of infinite-dimensional normalization. Classical methods based on the application of the theory of invariant integral manifolds and normal forms turn out to be directly inapplicable. Results. As the main results, special nonlinear boundary value problems are constructed, which play the role of normal forms. Their nonlocal dynamics determine the behavior of all solutions of the original system in the vicinity of the equilibrium state.