View source for Shadowing lemma

{{Short description|Lemma describing the behaviour of pseudo-orbits near a hyperbolic invariant set}}
In the [[Dynamical systems theory|theory of dynamical systems]], the '''shadowing lemma''' is a [[lemma (mathematics)|lemma]] describing the behaviour of pseudo-orbits near a [[Hyperbolic set|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<ref>{{MathWorld | title = Shadowing Theorem | urlname =  ShadowingTheorem }}</ref>) stays uniformly close to some true trajectory (with slightly altered initial position)—in other words, a pseudo-trajectory is "shadowed" by a true one.<ref>{{cite journal |last1=Hammel |first1=Stephan M |last2=Yorke |first2=James A |last3=Grebogi |first3=Celso |title=Numerical orbits of chaotic processes represent true orbits |journal=Bulletin of the American Mathematical Society |series=New Series |date=1988 |volume=19 |issue=2 |pages=465–469 |doi=10.1090/S0273-0979-1988-15701-1 |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-19/issue-2/Numerical-orbits-of-chaotic-processes-represent-true-orbits/bams/1183554729.full|doi-access=free }}</ref> 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.<ref>{{Cite journal |last1=Chandramoorthy |first1=Nisha |last2=Wang |first2=Qiqi |date=2021 |title=On the probability of finding nonphysical solutions through shadowing |url=https://linkinghub.elsevier.com/retrieve/pii/S0021999121002849 |journal=Journal of Computational Physics |language=en |volume=440 |pages=110389 |arxiv=2010.13768 |doi=10.1016/j.jcp.2021.110389|bibcode=2021JCoPh.44010389C |s2cid=225075706 }}</ref>

== Formal statement ==
Given a map ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''X'' of a [[metric space]] (''X'',&nbsp;''d'') to itself, define a '''&epsilon;-pseudo-orbit''' (or '''&epsilon;-orbit''') as a sequence <math>(x_n)</math> of points such that <math>x_{n+1}</math> belongs to a &epsilon;-neighborhood of <math>f(x_n)</math>.

Then, near a hyperbolic invariant set, the following statement holds:<ref>{{cite book |first1=A. |last1=Katok |first2=B. |last2=Hasselblatt |title=Introduction to the Modern Theory of Dynamical Systems |url=https://archive.org/details/introductiontomo0000kato |url-access=registration |location=Cambridge |publisher=Cambridge University Press |year=1995 |isbn=0-521-34187-6 |at=Theorem 18.1.2. }}</ref> 
Let &Lambda; be a hyperbolic invariant set of a [[diffeomorphism]] f. There exists a neighborhood U of &Lambda; with the following property: for any ''&delta;''&nbsp;>&nbsp;0 there exists ''&epsilon;''&nbsp;>&nbsp;0, such that any (finite or infinite) &epsilon;-pseudo-orbit that stays in U also stays in a &delta;-neighborhood of some true orbit.
:<math>
\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).
</math>

==See also==
* [[Chaotic systems]]
* [[Butterfly effect]]

== References ==
{{reflist}}

== External links ==
* [http://www.scholarpedia.org/article/Shadowing_lemma_for_flows Shadowing Theorem] on [[Scholarpedia]]
* [https://www.youtube.com/playlist?list=PLcqHTXprNMIPB6u3mUBWPBQyxHvqI1FXj Can a butterfly in Brazil control the climate of Texas?]

[[Category:Dynamical systems]]
[[Category:Lemmas]]


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