On the structure of the Hamiltonian phase flow near symmetric periodic solution

We consider an autonomous Hamiltonian system with two degrees of freedom, which is invariant under Klein four-group $K_4$ 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.