Dynamics of a system of two equations with a large delay

The local dynamics of systems of two equations with delay is considered. The main assumption is that the delay parameter is large enough. Critical cases in the problem of the stability of the equilibrium state are highlighted and it is shown that they have infinite dimension. Methods of infinite-dimensional normalisation were used and further developed. The main result is the construction of special nonlinear boundary value problems which play the role of normal forms. Their nonlocal dynamics determines the behaviour of all solutions of the original system in а neighbourhood of the equilibrium state.