Abstract
Numerical and analytical studies are performed for a model of three vibrational modes coupled by two Fermi resonances, in order to investigate its transition towards classical and quantum chaos. The major difficulty arises from the fact that there remains a classical conserved quantity and a good quantum number even in the chaotic regime. It is shown that, in spite of the conserved quantity, all the criteria for the appearance of classical local chaos (Lyapunov exponents, power spectra and Poincaré surfaces of section) give coherent information. The first steps of the transition towards chaos are analysed analytically in power spectra. The situation looks very different for quantum spectra. The remaining good quantum number causes the nearest neighbor distribution (NND) and the two-level correlation coefficient C to become almost insensible to the transition towards quantum chaos. It is shown that the fingerprint of the onset of chaos can however still be found in the plot of the number variance ∑2(L), where the second Fermi resonance causes minima to appear at values of L corresponding to the average distance between pairs of levels coupled by the resonance.
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