Time-dependent normal form Hamiltonian for dynamical equilibria

A new time-dependent normal-form procedure for dynamical equilibria (undergoing parametric excitation) of one-dimensional (1D) Hamiltonian systems is developed with the method of Lie transforms. The expansion is based on the Lewis invariant for the linearized motion. The time-dependent Hamiltonian normal form reduces smoothly to the usual representation in the autonomous limit. Illustrative examples of the formalism are focused on time-periodic systems and the dynamics of Hamiltonian switching.