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Guo Cheng Yuan and Brian R. Hunt

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Nonlinearity 12 (1999), 1207-1224.
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Online abstract and download information

Given a dynamical system and a function $f$ from the state space to the
real numbers, an optimal orbit for $f$ is an orbit over which the time average
of $f$ is maximal. In this paper we consider some basic mathematical
properties of optimal orbits: existence, sensitivity to
perturbations of $f$, and approximability by periodic orbits with low
period. For hyperbolic systems, we conjecture that for (topologically) generic
smooth functions, there exists an optimal periodic
orbit. In support of this conjecture, we prove that optimal periodic
orbits are insensitive to small $C^1$ perturbations of $f$, while
the optimality of a non-periodic orbit can be destroyed by arbitrarily
small $C^1$ perturbations.
In case there is no optimal periodic orbit for a given $f$, we
discuss the question of how fast the maximum average over orbits of
period at most $p$ must converge to the optimal average, as $p$
increases.

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*Updated: June 17, 1999*