Optimal periodic orbits of chaotic systems occur at low period

Brian R. Hunt and Edward Ott
Phys. Rev. E 54, 328 – Published 1 July 1996
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Abstract

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 that 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 γ, then the Lebesgue measure in γ 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. © 1996 The American Physical Society.

  • Received 31 January 1996

DOI:https://doi.org/10.1103/PhysRevE.54.328

©1996 American Physical Society

Authors & Affiliations

Brian R. Hunt and Edward Ott

  • Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742 Institute for Plasma Research, Institute for Systems Research, and Departments of Electrical Engineering and of Physics, University of Maryland, College Park, Maryland 20742

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Vol. 54, Iss. 1 — July 1996

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