On Finding Generalized Normal Forms of Systems with a Hamiltonian Unperturbed Part Using Belitskii’s Method

The study is a continuation of an article written in collaboration with V.V. Basov regarding finding the structures of generalized normal forms of two-dimensional autonomous systems of ordinary differential equations with a Hamiltonian unperturbed part and a non-Hamiltonian perturbation. The present article considers the case of systems of four and more equations with a Hamiltonian quasi-homogeneous unperturbed part generated by a Hamiltonian of form H = $$\sum\nolimits_{i = 1}^n {{H_i}({x_i},{y_i})} $$ . A special decomposition of the perturbation, similar to the decomposition into Hamiltonian and non-Hamiltonian components used by A. Baider and J. Sanders for the two-dimensional case, is used to reduce the problem of finding the generalized normal form of the considered system to the problem of the normal form of power series. Normalization of power series has been previously studied by G.R. Belitskii. The presented approach is based on Belitskii’s method and develops his ideas by introducing the notions of Hamiltonian resonance sets and truncated resonance sets; these terms are used to formulate the normalization theorem. As a corollary of the theorem, a generalization of the Takens result on the normal form of a system with a nilpotent linear part to the case of an arbitrary number of Jordan blocks is also presented. In particular, for n = 2, one of the generalized normal forms as defined by V.V. Basov for a system of four equations with an unperturbed part (y1, 0, y2, 0) is explicitly derived.