Discrete symmetric dynamical systems at the main resonances with application to axi-symmetric galaxies

A study of two-degrees-of-freedom systems with a potential which is discrete-symmetric (even in one of the position variables) is carried out for the resonance cases 1:2, 1:1, 2:1 and 1:3. To produce both qualitative and quantitative results, we obtain in each resonance case normal forms by higher order averaging procedures. This method is related to Birkhoff normalization and provides us with rigorous asymptotic estimates for the approximate solutions. The normal forms have been used to obtain a classification of possible local and global bifurcations for these dynamical systems. One of the applications here is to describe the two-parameter family of bifurcations obtained by detuning a one-parameter family studied by Braun. In all the resonances discussed an approximate integral of the motion other than the total energy exists, but in the 2:1 and 1:3 resonance cases this degenerates into the partial energy of the z motion. In conclusion some remarks are made on the relation between two-degrees-of-freedom systems and solutions of the collisionless Boltzmann equation. Moreover we are able to make some observations on the Henon-Heiles problem and certain classical examples of potentials.