Normalization flow

In the talk we consider a new approach to the theory of normal forms of the ODE system in the neighborhood of the equilibrium position. Traditional procedures perform normalization stepwise: the non-resonant terms in the Taylor expansion of the vector field turn to zero first in degree 2, then (by another substitution of variables) in degree 3, etc. We propose to proceed in a different way. Let us consider an infinite-dimensional space of all vector fields with a special point (equilibrium position) at zero. In this space, we construct a flow (generated by some differential equation) with the following properties. Shifts along the trajectories of this flow correspond to substitutions of variables. The flow moves towards the subspace of normal forms. Thus, the normalization procedure becomes continuous. The formal aspect of the theory (as in the traditional approach) is straightforward. The analytical aspect and the problems of convergence of series are, as usual, nontrivial.