Preprint des Projektes: SFB 343: Diskrete Strukturen in der Mathematik

Theorie und Numerik von Aufgaben der linearen Algebra und diskreter dynamischer Systeme

95-097 Yong-Kui Zou, Wolf-Jürgen Beyn.
Invariant Manifolds for Non-autonomous Systems with Application to One-step Methods

In this paper we study the existence of invariant manifolds for a special type of nonautonomous systems which arise in the study of discretization methods. According to [10], a one-step scheme of step-size h for an autonomous system can be interpreted as the h-flow of a perturbed nonautonomous system. The perturbation is `rapidly forced' in the sense that it is periodic with respect to time with period h. Assuming a saddle node for the autonomous system, we prove that these rapidly forced perturbations have center manifolds which exist in a uniform neighborhood and which converge to a center manifold of the autonomous system as h tends to zero. Our results are applied to obtain a smooth continuation as well as estimates of the well known center manifolds for one-step schemes. They also form the basis for studying saddle-node homoclinic orbits under discretization.