Shadowing lemma

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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[edit]

Given a map f : X → X of a metric space (Xd) 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[edit]

  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.