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.

A Shadowing lemma is also a fictional creature in the Discworld. See Shadowing lemma.

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 ${\displaystyle (x_{n})}$ of points such that ${\displaystyle x_{n+1}}$ belongs to a ε-neighborhood of ${\displaystyle f(x_{n})}$.

Then, near a hyperbolic invariant set, the following statement holds:[2] 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.

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

See also

• Butterfly effect - sensitive dependence on initial conditions

References

1. Weisstein, Eric W. "Shadowing Theorem". MathWorld.
2. Katok, A.; Hasselblatt, B. (1995). Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press. Theorem 18.1.2. ISBN 0-521-34187-6.

• Scholarpedia article Shadowing Theorem