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Brian R. Hunt and Edward Ott

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Phys. Rev. E 54 (1996), 328-337.
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Invariant sets embedded in a chaotic attractor can generate time
averages that differ from the average generated by typical orbits on
the attractor. Motivated by two different topics (namely, controlling
chaos and riddled basins of attraction), we consider the question of
which invariant set yields the largest (optimal) value of an average
of a given smooth function of the system state. We present numerical
evidence and analysis which indicate that the optimal average is
typically achieved by a low period unstable periodic orbit embedded in
the chaotic attractor. In particular, our results indicate that, if
we consider that the function to be optimized depends on a parameter
$\gamma$, then the Lebesgue measure in $\gamma$ corresponding to
optimal periodic orbits of period $p$ or greater decreases
exponentially with increasing $p$. Furthermore, the set of parameter
values for which optimal orbits are nonperiodic typically has zero
Lebesgue measure.

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*Last updated: May 31, 1998*