Shadowing lemma
In the theory of dynamical systems, the shadowing lemma is a lemma describing the behaviour of pseudo-orbits near a hyperbolic invariant set. Informally, the theory states that every pseudo-orbit stays uniformly close to some true trajectory —in other words, a pseudo-trajectory is "shadowed" by a true one. This suggests that numerical solutions can be trusted to represent the orbits of the dynamical system. However, caution should be exercised as some shadowing trajectories may not always be physically realizable.
Formal statement
Given a map f : X → X of a metric space to itself, define a ε-pseudo-orbit as a sequence of points such that belongs to a ε-neighborhood of.Then, near a hyperbolic invariant set, the following statement holds:
Let Λ be a hyperbolic invariant set of a diffeomorphism f. There exists a neighborhood U of Λ with the following property: for any δ > 0 there exists ε > 0, such that any ε-pseudo-orbit that stays in U also stays in a δ-neighborhood of some true orbit.