Equivariant singularity theory with distinguished parameters: Two case studies of resonant Hamiltonian systems

We consider Hamiltonian systems near equilibrium that can be (formally) reduced to one degree of freedom. Spatiotemporal symmetries play a key role. The planar reduction is studied by equivariant singularity theory with distinguished parameters. The method is illustrated on the conservative spring-pendulum system near resonance, where it leads to integrable approximations of the iso-energetic Poincare map. The novelty of our approach is that we obtain information on the whole dynamics, regarding the (quasi-) periodic solutions, the global configuration of their invariant manifolds, and bifurcations of these.