Approximate and real trajectories for generic dynamical systems
The authors show that generic dynamical systems have a weak shadowing property. In particular, they show that, for a generic dynamical system \(\varphi\) in the space of discrete continuous dynamical systems on a compact manifold with the usual \(C^ 0\) topology, given \(\varepsilon > 0\) there exists \(\delta > 0\) such that \(\delta\)-trajectories of \(\varphi\) are \(\varepsilon\)-close to real trajectories. A \(\delta\)-trajectory is an idealization of the notion of a ``locally accurate'' numerical approximation to a real trajectory. The shadowing result in this paper involves a weaker property than the pseudo-orbit tracing property (POTP). The authors show only that \(\delta\)-trajectories of generic dynamical systems are ``weakly \(\varepsilon\)-traced'' by true trajectories. However, there are no dimensional restrictions on this result. The authors also present some results on the inverse problem of tracing real trajectories by \(\delta\)-trajectories.
numerical solutions of differential equations
scientific article; zbMATH DE number 743909