On finding generalized normal forms of systems with a Hamiltonian unperturbed part using the Belitskii method

Abstract

This work continues an article written in collaboration with V. V. Basov about finding the structures of generalized normal forms of two-dimensional autonomous systems of ordinary differential equations with a Hamiltonian unperturbed part and non-Hamiltonian perturbation. In this article we consider the case of systems of four and more equations with a Hamiltonian quasi-homogeneous unperturbed part generated by the Hamiltonian of the form H = n i=1 Hi(xi, yi). By means of a special decomposition of the perturbation similar to splitting into Hamiltonian and non-Hamiltonian components that was used by A. Baider and J. Sanders in two-dimensional case, we reduce the problem of finding the generalized normal form for such a system to the question about the normal form of power series. Normalization of a power series was studied earlier by G. R. Belitskii. We use the method of G. R. Belitskii and develop his ideas by introducing the notions of Hamiltonian resonant and reduced resonant sets, and in these terms formulate the normalization theorem. As a consequence of the theorem we give a generalization of the result due to Takens on the normal form of a system with a nilpotent linear part to the case of an arbitrary number of Jordan blocks. In particular, for n = 2 we find explicitly one of the generalized normal forms in the sense of V. V. Basov for a system of four equations with an unperturbed part (y1, 0, y2, 0). Refs 7.