Shadowing lemma
Shadowing lemma (short and clear)
In dynamical systems, the shadowing lemma explains how approximate trajectories behave near a hyperbolic invariant set. Suppose we have a map f: X → X on a metric space (X, d). An ε-pseudo-orbit is a sequence (x_n) where each next point x_{n+1} is within ε of f(x_n).
If Λ is a hyperbolic invariant set for f, there is a neighborhood U of Λ with this property: for every δ > 0 there exists an ε > 0 such that any ε-pseudo-orbit that stays inside U (for all time) also stays within δ of some true orbit of f. In other words, every small-error path near Λ can be closely matched by a real trajectory.
This means numerical simulations, which produce pseudo-orbits with small step errors, can often be trusted to reflect actual behaviors near Λ. But be careful: not every shadowing path is guaranteed to be physically realizable.
This page was last edited on 2 February 2026, at 08:13 (CET).