Periodic orbits emanating from a resonant equilibrium

For a conservative Hamiltonian system with two degrees of freedom, in the case where the two frequencies at an equilibrium of the elliptic type are commensurable or close to being so, completely canonical transformations can be formally constructed in explicit terms under the form of Lie transforms to the effect that it renders one angle coordinate ignorable and gives to the transformed Hamiltonian the form of what Garfinkel calls an ideal problem of resonance. For the problem so reduced, the unnormalized residual being omitted, natural families of periodic orbits are analyzed, their emergence from the equilibrium is discussed as well as their characteristic exponents. Special attention is given to the evolution of the system of natural families under a continuous transition through the resonance band.