First we remind the normal form near a stationary solution of an autonomous Hamiltonian system. Second we consider the linear periodic Hamiltonian systems. For them we find normal forms of Hamiltonian functions in both complex and real cases. The real case has a specificy in the case of parametric resonance.Then we find normal forms of the Hamiltonian functions for nonlinear periodic systems. By means of additional canonical transformation of coordinates, such system always is reduced to an autonomous Hamiltonian system, which preserves all small parameters and symmetries of the initial system. Its local families of stationary points correspond to families of periodic solutions of the initial system. We make similar theory in vicinity of periodic solution to an autonomous system.

Hamiltonian system, complex normal form, real normal form, reduced normal form, parametric resonance

Mathematical modelling in actual problems of science and technics

Bruno Alexander Dmitrievich,  ,  orcid.org/0000-0002-7465-1258,  KIAM RAS