The bifurcation structure is investigated for a Hamiltonian for the three coupled nonlinear vibrations of a highly excited triatomic molecule. The starting point is a quantum Hamiltonian used to fit experimental spectra. This Hamiltonian includes 1:1 Darling-Dennison resonance coupling between the stretches, and 2:1 Fermi resonance coupling between the stretches and bend : A classical Hamiltonian is obtained using the Heisenberg correspondence principle. Surfaces of Section show a pronounced degree of chaos at high energies, with a mixture of chaotic and regular dynamics. The large-scale bifurcation structure is found semianalytically; without recourse to numerical solution of Hamilton’s equations, by taking advantage of the fact that the spectroscopic Hamiltonian has a conserved polyad quantum number, corresponding to an approximate constant of the motion of the molecule. Bifurcation diagrams are analyzed for a number of molecules including H2O, D2O, NO2, ClO2, O-3, and H2S.