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Approximate and real trajectories for generic dynamical systems. (English) Zbl 0821.58036
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.

MSC:
37-XX Dynamical systems and ergodic theory
65J99 Numerical analysis in abstract spaces
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