Resonant elliptic equilibria in Hamiltonian systems

The main focus of this thesis is the dynamics of a two-degree-of-freedom Hamiltonian system near an elliptic equilibrium point in 1 : ±2 resonance, as described by its integrable resonant normal form approximation. After applying singular reduction, the system is studied on the reduced phase space that Kummer called the "orbit manifold". We give a complete description of the critical values of the energy–momentum mapping which than enables us to study the topology of the regular, and more importantly, the singular fibres. We then derive equations for the period of the reduced system, for the rotation number and the non-trivial action. Although some of these results are not new, our approach does not rely on compact fibration and is based on a similar approach introduced earlier by S. Vũ Ngoc for focus–focus points. The non-trivial action of the system enables us to establish fractional monodromy very elegantly by deriving the transition matrix for the actions directly. Both the isoenergetic non-degeneracy condition and the Kolmogorov non-degeneracy condition of KAM theory are derived and analysed for the resonant case. It tuns out that the twist vanishes in a neighbourhood of the equilibrium point for the sign-indefinite case. The Kolmogorov condition, however, is always satisfied near 1 : ±2 resonant equilibria. [Continues.]