Poincaré's theorem and subdynamics for driven systems

Large dynamical systems with explicit time dependence will be studied. For these systems (which we call driven systems), there exist resonances between internal frequencies and/or between internal and external frequencies. As in the time-independent case, the usual canonical or unitary perturbation theory leads to divergences due to the resonances. As a result, there exist no trajectories analytic in both the coupling constant and the initial data. This is a generalization of Poincaré's theorem on non-integrability and extends the notion of large Poincaré systems (LPS), i.e., systems with a continuous spectrum and a continuous set of resonances. Here the resonances involve external frequencies. Along the line of the subdynamics theory developed by Prigogine and his co-workers, we study LPS within the Liouville-space formalism. We construct projection operators which decompose the equations of motion and are analytic in the coupling constant. Our approach recovers Coveney's theory of time-dependent subdynamics but is based on recursion formulas, which significantly simplify the construction of the projection operators. These projectors are non-Hermitian and provide a description with broken time symmetry. As an application, we study the modification of the well-known three stages of the decay process from a time-dependent perturbation.