Characterization of the natural measure by unstable periodic orbits in nonhyperbolic chaotic systems

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Abstract

The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of all unstable periodic orbits embedded in part of the chaotic set contained in that region. This result has been rigorously shown to be valid for hyperbolic chaotic systems. Chaotic sets encountered in most physical situations, however, are typically nonhyperbolic. The purpose of this paper is to test the goodness of the unstable periodic-orbit characterization of the natural measure for nonhyperbolic chaotic systems. We first directly compare the natural measure from a typical trajectory on the chaotic set with that evaluated from unstable periodic orbits embedded in the set. As an indirect check, we then compute the difference between the long-time average values of physical quantities evaluated with respect to a typical trajectory and those computed from unstable periodic orbits. Results with the Hénon map for which periodic orbits can be enumerated lend credence to the conjecture that the unstable periodic-orbit theory of the natural measure is applicable to nonhyperbolic chaotic systems.

Original languageEnglish (US)
Pages (from-to)6531-6539
Number of pages9
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume56
Issue number6
StatePublished - 1997
Externally publishedYes

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Chaotic System
Periodic Orbits
Unstable
orbits
trajectories
Trajectory
Time-average
Hyperbolic Systems
Phase Space
Valid

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Condensed Matter Physics
  • Statistical and Nonlinear Physics

Cite this

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title = "Characterization of the natural measure by unstable periodic orbits in nonhyperbolic chaotic systems",
abstract = "The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of all unstable periodic orbits embedded in part of the chaotic set contained in that region. This result has been rigorously shown to be valid for hyperbolic chaotic systems. Chaotic sets encountered in most physical situations, however, are typically nonhyperbolic. The purpose of this paper is to test the goodness of the unstable periodic-orbit characterization of the natural measure for nonhyperbolic chaotic systems. We first directly compare the natural measure from a typical trajectory on the chaotic set with that evaluated from unstable periodic orbits embedded in the set. As an indirect check, we then compute the difference between the long-time average values of physical quantities evaluated with respect to a typical trajectory and those computed from unstable periodic orbits. Results with the H{\'e}non map for which periodic orbits can be enumerated lend credence to the conjecture that the unstable periodic-orbit theory of the natural measure is applicable to nonhyperbolic chaotic systems.",
author = "Ying-Cheng Lai",
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AU - Lai, Ying-Cheng

PY - 1997

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AB - The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of all unstable periodic orbits embedded in part of the chaotic set contained in that region. This result has been rigorously shown to be valid for hyperbolic chaotic systems. Chaotic sets encountered in most physical situations, however, are typically nonhyperbolic. The purpose of this paper is to test the goodness of the unstable periodic-orbit characterization of the natural measure for nonhyperbolic chaotic systems. We first directly compare the natural measure from a typical trajectory on the chaotic set with that evaluated from unstable periodic orbits embedded in the set. As an indirect check, we then compute the difference between the long-time average values of physical quantities evaluated with respect to a typical trajectory and those computed from unstable periodic orbits. Results with the Hénon map for which periodic orbits can be enumerated lend credence to the conjecture that the unstable periodic-orbit theory of the natural measure is applicable to nonhyperbolic chaotic systems.

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