The 1:1 resonance in Hamiltonian systems

Abstract

Two-degree-of-freedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are well understood for most resonances k:l, but not the semisimple cases 1:1 and 1:-1. A two-degree-of-freedom Hamiltonian system can be approximated to any order by an integrable approximation. The reason is that the normal form of a Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. Thus we study $S^1$--symmetric systems. The question we wish to address is about the co-dimension of such a system in 1:1 resonance with respect to left-right-equivalence, where the right action is $S^1$--equivariant. The result is a co-dimension five unfolding of the central singularity. Two of the unfolding parameters are moduli and the remaining non-modal parameters are the ones found in the linear unfolding of this system.