## 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.