Catastrophe map classification of the generalized normal–local transition in Fermi resonance spectra

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Abstract
Catastrophe theory is used to classify the dynamics of spectra of resonantly coupled vibrations, based on earlier work on the bifurcation structure of the Darling–Dennison and 2:1 Fermi resonance fitting Hamiltonians. The goal is a generalization of the language of the ‘‘normal–local transition’’ to analyze experimental spectra of general resonant systems. The set of all fixed points of the Hamiltonian on the polyad phase sphere for all possible molecular parameters constitutes the catastrophe manifold. The projection of this manifold onto the subspace of molecular parameters is the catastrophe map. The map is divided into zones; each zone has its own characteristic phase sphere structure. The taxonomy of global phase sphere structures within all zones gives the classification of the semiclassical dynamics. The 1:1 system, with normal–local transition, is characterized by cusp catastrophes, with elementary pitchfork bifurcations. In contrast, the 2:1 system is characterized by fold catastrophes, with elementary transcritical bifurcations. The catastrophe map can be used in a new method to classify experimental spectra on the basis of the system’s underlying semiclassical dynamics. The catastrophe map classification appears to persist for nonintegrable, chaotic Hamiltonians, indicating the utility of catastrophe theory for understanding the morphology of chaotic systems.