The unfolding theory, to be developed below, thenThe conjugate pair of imaginary eigenvalues gives rise to a formal rotational symmetry in all unfoldings, i.e. a rotational symmetry in their Taylor series. Here the series is considered in dependence on both the phase space variables and the parameters. This is an application of Normal Form Theory, where the terms of the formal power series are changed by canonical coordinate transformations in an inductive process. The symmetry, thus obtained, enables a formal reduction to 1 degree of freedom, around an equilibrium point with a double eigenvalue 0. For similar approaches see some of the above references, also compare e.g. Arnold [6], Takens [43,44], Broer [10,11], Golubitsky and Stewart [23] and Broer and Vegter [16]. This Normal Form Theory will be presented in Section 3.In Section 2, we begin studying the 1 degree of freedom 'backbone'-problem in its own right. Since in the plane the integral curves of the systems are the level sets of the Hamiltonian functions, here the problem reduces to Singularity Theory, e.g. In Section 4 the connection is made between the planar reduction of the symmetric system and the 'backbone'-system of Section 2. It turns out that the formal integral, obtained by normalization, is a distinguished parameter in the sense of Golubitsky and Schaeffer [51] and Schecter [41], also compare Wassermann [52]. Now Singularity Theory yields new normal forms, that are polynomial of degree 3, at least in the phase-space variables. We note that here the normalizing transformations no longer need to be canonical. Technical details from Singularity Theory have been collected in an appendix (Section 7). After this we suspend, or dereduce, to the original 4-dimensional setting, so obtaining an integrable, i.e. rotationally symmetric, approximation of the original unfolding.Thus we obtain a perturbation problem, similar to e.g. Broer [10,12] or Braaksma and Broer [8]. The perturbation term is of arbitrarily high order, both in the phase space variables and the parameters. This problem is briefly addressed in Section 5.Remarks. (i) Although in the original family only generic restrictions are imposed on the lower order terms, by Normal Form Theory and Singularity Theory, the corresponding unfolding is reduced to an arbitrarily flat perturbation of a normal form, completely determined by a 1 degree of freedom system, which is polynomial of degree 3. This polynomial character Vol. 44, 1993 A normally elliptic Hamiltonian bifurcation 391 may explain why certain 'integrable' characteristics in the unfoldings are so persistent. For a related comment we refer to Verhulst [49].(ii) Our approach differs from e.g. Van der Meer [46], also compare Duistermaat [19]. In [46,19] the dynamics is reduced by the energy-momentum map as well as the Moser-Weinstein method. This gives information related to specific periodic solutions, that is also important for the dynamics as a whole. Instead, we just factor out a formal rotational symmetry, and so seem to get a more ...

Eigenvalues and wave functions for acetaldehyde in its electronic ground state and the (n,3s) Rydberg state are calculated using the rho-axis method in combination with the high barrier approximation. The rotational structure of the first members of the torsional sequence in the two-photon spectrum is analyzed in terms of a zeroth-order rotation–torsional Hamiltonian. Combining experimental information on band positions and rotational contours, we determine the range of internal rotation constants and barrier heights, which is compatible with the experimental data. The analysis yields the following parameters: F=8.45±0.15 cm−1, V3=809±10 cm−1, ρ=0.33±0.03, ΘRAM=4.5°±0.8°, and APA=1.990 cm−1, BPA=0.330 cm−1, CPA=0.298 cm−1.

No abstract