The normal and the scanning normal form of nonlinear dynamical systems

Abstract

In this paper a new analytical technique for constructing a normalizing change of variables is presented. The Generalized Normal Form Method allows one to widen substantially the class of dynamic systems suitable for normalization and to include new kinds of dynamic systems in the method scheme which have never been considered previously. The procedure described reveals clearly the similarity and difference between the traditional local approximation of a small parameter and modern methods based on the change of variables. The Normal Form Method allows the maximum possible simplification of the vector field in the neighborhood of a singular point of the phase space. However, under such a simplification the normalized vector fields at various points are not smoothly consistent. The construction of the Scanning Normal Form (SNF) allows one to get over this discrepancy. The SNF is obtained in an open neighborhood of an arbitrary fixed point of the phase space, so that both the vector field determined by the SNF and normalizing change of variables depend smoothly on the coordinates of this point. The smooth normalization of the vector field in the extended parametric family whose dimension depends on the dimension of the phase space leads formally to the SNF. The SNF method leads to continuous approximation of the solutions of dynamic systems and can be considered as a basis of parallel computing procedures for the large-scale nonlinear dynamic systems.