[Here is an article](http://www.pnas.org/content/99/22/13975.full) on Hamilton systems near strongly resonant periodic orbits. From the first page:

> In a Hamiltonian system periodic orbits are not usually isolated but form one-parametric families. Naturally the value of the Hamiltonian function H plays the role of the parameter. Thus even in the case when the original Hamiltonian does not explicitly contain any parameter, it is possible to observe bifurcations of periodic orbits. A bifurcation corresponds to a resonance between the frequency of a periodic orbit and the frequency of small oscillations around it. In a generic situation there is a family of hyperbolic periodic orbits of a multiple period, which shrinks to the resonant periodic orbit at an exact resonance. Separatrices of the hyperbolic periodic orbit have to intersect due to Hamiltonian nature of the problem. Segments of separatrices of the corresponding resonant normal form make up a closed loop around the periodic trajectory. In section E, appendix 7 of ref. 1, Arnold pointed out that there should be an important qualitative difference between the original Hamiltonian system and its normal form due to the splitting of separatrices.