First 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. Then we find normal forms of the Hamiltonian functions for nonlinear periodic systems also in complex and real cases. 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. All that concludes the study of the problem mentioned in the title and partially given in Ch. II of the book A.D. Bruno “The Restricted 3-Body Problem”. Berlin. Walter de Grouter, 1994. We consider a nontrivial example with two degrees of freedom.

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

Mathematical modelling in actual problems of science and technics

Bruno Alexander Dmitrievich,  abruno@keldysh.ru,  orcid.org/0000-0002-7465-1258,  KIAM RAS