Pugh's closing lemma
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:
- Let
be 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.