Phase space bifurcation structure and the generalized local-to-normal transition in resonantly coupled vibrations

Abstract

The generalization of the local-to-normal transition seen in symmetric triatomics is considered for nonsymmetric molecules and 2:1 Fermi resonance systems. A straightforward generalization based on a division of phase space into local and normal regions is not possible. Instead, classification of the phase space bifurcation structure is presented as the complete generalization of the local–normal concept for all spectroscopically relevant systems of two vibrations interacting via a single nonlinear resonance. The polyad phase sphere (PPS) is shown to be the natural arena to analyze the bifurcation structure for resonances of arbitrary order. For 1:1 and 2:1 resonances, the bifurcation problem is reduced to one or two great circles on the phase sphere. All bifurcations are shown to be examples of elementary bifurcations of vector fields in one dimension. The classification of the bifurcation structure is therefore governed and greatly simplified by the theory of the universal unfolding and codimension of elementary bifurcations. The implications for large-scale bifurcation structure and transport in molecules with chaotic motion are briefly discussed.