An energy–momentum map for the time-reversal symmetric 1:1 resonance with Z2×Z2Z2×Z2 symmetry

• Analysis of the bifurcation sequences of symmetric 1:1 resonant Hamiltonian normal forms.
• Catastrophe map illustrating global properties of the system.
• Energy–momentum map with bifurcation thresholds and orbit families.
• Actions, periods and rotation numbers.

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 Z2×Z2Z2×Z2 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.