Periodic Perturbations: Parametric Systems

We are not going to present the classical results on linear parametric systems, since they are widely discussed in literature. Instead, we shall consider nonlinear parametric systems and discuss the conditions of new motion existence in the resonance zones: the regular ones (on an invariant torus) and the irregular ones (on a quasi-attractor). On the basis of the self-oscillatory shortened system which determines the topology of resonance zones, we study the transition from a resonance to a non-resonance case under a change of the detuning. We then apply our results to some concrete examples. It is interesting to study the behavior of a parametric system when the ring-like resonance zone is contracted into a point, i.e., to describe the bifurcations which occur in the course of transition from the plain nonlinear resonance to the parametric one. We are based on article, and we follow a material from the book.