Higher Order Resonance in Two Degree of Freedom Hamiltonian System

This paper reviews higher order resonance in two degrees of freedom Hamiltonian systems. We consider a positive semi-definite Hamiltonian around the origin. Using normal form theory, we give an estimate of the size of the domain where interesting dynamics takes place, which is an improvement of the one previously known. Using a geometric numerical integration approach, we investigate this in the elastic pendulum to find additional evidence that our estimate is sharp. In an extreme case of higher order resonance, we show that phase interaction between the degrees of freedom occurs on a short time-scale, although there is no energy interchange.