The simplest critical cases in the dynamics of nonlinear systems with small diffusion

Systems of nonlinear equations of parabolic type provide models for many processes and phenomena. A special role is played by systems with relatively small diffusion coefficients. In investigating the dynamical properties of solutions, the diffusion coefficients being small leads to the appearance of infinite-dimensional critical cases in problems on the stability of solutions. In this paper we study the simplest and most important of these critical cases. Special nonlinear evolution equations are constructed which play the role of normal forms; their nonlocal dynamics determines the behaviour of solutions of the original system in a small neighbourhood of an equilibrium state. The importance of the renormalization procedure is demonstrated.