We consider an autonomous Hamiltonian system with two degrees of freedom, which is invariant under Klein four-group K4 of linear canonical automorphisms of the extended phase space of the system. The sequence of symplectic transformations of monodromy matrix of a symmetric periodic solution is proposed. Three types of bifurcations of a family of symmetric periodic solutions – saddlenode bifurcation, pitch-fork bifurcation and period multiplying bifurcation – are investigated by means of these transformations. For last two types of bifurcations different scenarios are shown for the case of doubly symmetric periodic solutions of the Hill problem.

periodic solution, symmetry, monodromy matrix, Hill problem, bifurcation of periodic solution

Mathematical modelling in actual problems of science and technics

Batkhin Alexander Borisovich,  batkhin@gmail.com,  orcid.org/0000-0001-8871-4697,  KIAM RAS