Bifurcations of Periodic Solutions of a Hamiltonian System with a Discrete Symmetry Group

Abstract

An autonomous Hamiltonian system with two degrees of freedom that is invariant under the Klein four-group of linear canonical automorphisms of the extended phase space of the system is studied. A sequence of symplectic transformations of the monodromy matrix of the symmetric periodic solution of this system is constructed. Using these transformations, the structure and bifurcation of the phase flow in the vicinity of this solution is investigated. It is shown that the bifurcations corresponding to the multiple period increase are different for the solutions with double symmetry and the solutions with a single symmetry. An example of critical periodic solutions of the family of doubly symmetric orbits of the plane circular Hill problem is discussed. The majority of tedious analytical calculations are performed using packages for the computation of Grobner bases and for the work with polynomial ideals in the computer algebra system Maple.