Shadowing lemma

A Shadowing lemma is also a fictional creature in the Discworld. See 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 (which one can think of as a numerically computed trajectory with rounding errors on every step) stays uniformly close to some true trajectory (with slightly altered initial position) — in other words, a pseudo-trajectory is "shadowed" by a true one. Incapability of the shadowing lemma on digital chaos are presented in the International Journal of Bifurcation and Chaos, Sec. 2.2.3.

Formal statement

Given a map f : X → X of a metric space (X, d) to itself, define a ε-pseudo-orbit (or ε-orbit) as a sequence $(x_n)$ of points such that $x_{n+1}$ belongs to a ε-neighborhood of $f(x_n)$.

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 (finite or infinite) ε-pseudo-orbit that stays in U also stays in a δ-neighborhood of some true orbit.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \forall (x_n),\, x_n\in U, \, d(x_{n+1},f(x_n))<\varepsilon \quad \exists (y_n), \, \, y_{n+1}=f(y_n),\quad \text{such that} \,\, \forall n \,\, x_n\in U_{\delta}(y_n).

References

• Scholarpedia article Shadowing Theorem