View source for Shadowing lemma for flows

==Chaos==
 
Shadowing has many applications in the theory of dynamical systems especially
to the existence of topological conjugacies (Kuznetsov (2007)). An application to chaos is described here.
In the flow context, the analogous theorem to Smale's (Smale and Shub (2007)) is that of Sil'nikov, which describes the chaotic dynamics in the neighbourhood of a transversal
homoclinic orbit associated with a hyperbolic periodic orbit of a flow.
One can apply Smale's theorem to an appropriately 
defined return map to verify chaotic behaviour in the neighbourhood of 
such a transversal periodic-to-periodic homoclinic orbit. However we can also use the 
discrete shadowing theorem above to verify the chaotic dynamics. Corresponding to a sequence of symbols, a pseudo orbit is constructed and then the shadowing lemma is used to
prove the existence of a nearby true orbit. In this fashion we can describe all 
the orbits which stay in a neighbourhood of the homoclinic orbit (Palmer (2000)).