- Interpretation
- See also
- References
In mathematics, Pugh's closing lemma is a result that links periodic orbit solutions of differential equations to chaotic behaviour. It can be formally stated as follows:
Letbe a
diffeomorphism of a compact smooth manifold
. Given a nonwandering point
of
, there exists a diffeomorphism
arbitrarily close to
such that
Interpretation
Pugh's closing lemma means, for example, that any chaotic set in a bounded continuous dynamical system corresponds to a periodic orbit in a different but closely related dynamical system. As such, an open set of conditions on a bounded continuous dynamical system that rules out periodic behaviour also implies that the system cannot behave chaotically; this is the basis of some autonomous convergence theorems.
See also
- Smale's problems
References
1. ^{{cite journal |first=Charles C. |last=Pugh |title=An Improved Closing Lemma and a General Density Theorem |journal=American Journal of Mathematics |volume=89 |issue=4 |pages=1010–1021 |year=1967 |doi=10.2307/2373414 }}
{{PlanetMath attribution|id=5526|title=Pugh's closing lemma}} 3 : Dynamical systems|Lemmas|Limit sets
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