| It was proved by Anosov and Bowen that approximate orbits (or pseudo orbits) of chaotic dynamical systems are shadowed by true orbits.Since computed orbits of chaotic dynamical systems are finite
pseudo orbits, the ideas of Anosov and Bowen can be applied to them.It can be shown that
(i) computed orbits of chaotic dynamical systems can be shadowed for long times by true orbits, a result that seems surprising given that the true orbit which has the same initial condition as the
computed orbit diverges at an exponential rate from the computed orbit.
(ii) Shadowing techniques can be used to show that computed orbits of chaotic systems which appear to be periodic are near true periodic orbits; that this can be done is also surprising since such orbits are necessarily unstable and may be of very long period.
(iii) These methods can be also used to find and rigorously prove the existence of transversal homoclinic orbits in chaotic systems.Such orbits are interesting because Smale proved that a dynamical system is chaotic in the neighborhood of a transversal homoclinic orbit. Many allegedly chaotic systems have not been proved to be chaotic. Rigorously proving the existence of a transversal homoclinic orbit demonstrates that these systems contain a set on which the dynamics is chaotic.
|