Guocheng Yuan and Brian R Hunt 1999 Nonlinearity 12 1207 doi:10.1088/0951-7715/12/4/325
Optimal orbits of hyperbolic systems
Guocheng Yuan and Brian R Hunt
Show affiliationsRecommended by L Bunimovich
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 C1 perturbations of f, while the optimality of a non-periodic orbit can be destroyed by arbitrarily small C1 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.
- PACS
- MSC
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37D45 Strange attractors, chaotic dynamics
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
- Subjects
- Dates
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Issue 4 (July 1999)
Received 3 September 1998, in final form 28 April 1999
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Optimal orbits of hyperbolic systems
Guocheng Yuan and Brian R Hunt 1999 Nonlinearity 12 1207
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