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^[1]) 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,^[2] 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:^[3] 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.

$\forall (x_n),\, x_n\in U, \, d(x_{n+1},f(x_n))<\varepsilon \quad \exists (y_n), \, \, y_{n+1}=f(y_n),\quad \text{such that} \,\, \forall n \,\, x_n\in U_{\delta}(y_n).$

References

1. Weisstein, Eric W., "Shadowing Theorem", MathWorld.
2. Shujun Li, Guanrong Chen and Xuanqin Mou (2005). "On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps". International Journal of Bifurcation and Chaos 15 (10): 3119–3151. doi:10.1142/S0218127405014052.
3. A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Theorem 18.1.2.

• Scholarpedia article Shadowing Theorem