An energy–momentum map for the time-reversal symmetric 1:1 resonance with symmetry

We present a general analysis of the bifurcation sequences of periodic orbits in general position of a family of reversible 1:1 resonant Hamiltonian normal forms invariant under Z 2 × Z 2 symmetry. The rich structure of these classical systems is investigated both with a singularity theory approach and geometric methods. The geometric approach readily allows to find an energy–momentum map describing the phase space structure of each member of the family and a catastrophe map that captures its global features. Quadrature formulas for the actions, periods and rotation number are also provided.