Approximate and Real Trajectories for Generic Dynamical Systems

In this paper we show that generic dynamical systems have a weak shadowing property, namely that for a generic dynamical system φ, in the space of discrete continuous dynamical systems on a compact manifold M with the usual C0 topology, given ϵ > 0, there exists a δ > 0 such that δ-trajectories are ϵ-close to real trajectories. There are no dimensional restrictions on this result. We also give some results on the "inverse" problem of tracing real trajectories by δ-trajectories. These problems are motivated by questions regarding the validity of numerical solutions of differential equations