Bifurcations in Hamiltonian systems around symmetric resonances

We investigates the orbit structure of two degrees of freedom nonlinear Hamiltonian systems around symmetric resonances. With this we mean Hamiltonian dynamical systems close to an equilibrium, invariant with respect to reflection symmetries in both configuration variables, in addition to the time reversion symmetry, and with quadratic part with unperturbed frequencies close to a resonant ratio. Such systems are naturally apt to be treated as perturbed nonlinear oscillators. The motivation behind this work lies in the study of problems arising in Galactic Dynamics, in which reflection symmetries appear to describe the orbital structure of elliptical galaxies. The analysis is performed combining three different mathematical tools: Perturbation Theory (in particular, the Poincaré-Birkhoff normal forms), Singularity Theory and Lie Symmetry Theory. Moreover, the very nature of the problems which motivated the study leads to consider not only standard perturbation and normal forms theory, but also the Verhulst’s approach through “detuned” resonances. The most relevant resonances and related bifurcations are analyzed, providing quantitative predictions, in the form of energy threshold values, which determine the appearance of the main periodic orbits.