Pugh's closing lemma - Alchetron, The Free Social Encyclopedia

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 f : M → M be a C 1 diffeomorphism of a compact smooth manifold M . Given a nonwandering point x of f , there exists a diffeomorphism g arbitrarily close to f in the C 1 topology of Diff 1 ⁡ ( M ) such that x is a periodic point of g .

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.

References

Pugh's closing lemma Wikipedia

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