Chains with Diffusion-Type Couplings Containing a Large Delay

We investigate the local dynamics of a system of oscillators with a large number of elements and with diffusion-type couplings containing a large delay. We distinguish critical cases in the problem of stability of the zero equilibrium state and show that all of them have infinite dimensions. Using special methods of infinite normalization, we construct quasinormal forms, that is, nonlinear boundary value problems of parabolic type, whose nonlocal dynamics determines the behavior of solutions of the original system in a small neighborhood of the equilibrium state. These quasinormal forms contain either two or three spatial variables, which emphasizes the complexity of dynamic properties of the original problem.