Bifurcation analysis of higher m:n resonance spectroscopic Hamiltonian

The classical phase space structure of a spectroscopic Hamiltonian for two coupled vibrational modes is analyzed using bifurcation theory, classified on catastrophe maps, for a variety of higher order resonances (3:2, 4:2, 5:2, 6:2 and 4:4, 5:4, 6:4), cases not considered in previous work. A type of bifurcation not encountered for lower resonance orders, based on overlap of separatrices rather than change in behavior of fixed points, is analyzed, and a procedure is developed to augment the catastrophe map. Energy level patterns are associated with the new resonances, in analogy with the patterns of adjacent level spacings considered earlier for the 2:1 resonance.