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

Y C Lai, Ying-Cheng Lai

Research output: Contribution to journalArticle

14 Citations (Scopus)

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 Henon 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
Pages (from-to)6531-6539
Number of pages9
JournalPhysical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume56
Issue number6
Publication statusPublished - Dec 1997

Keywords

  • STRANGE ATTRACTORS
  • GENERALIZED DIMENSIONS
  • FRACTAL MEASURES
  • SINGULARITIES
  • TRAJECTORIES
  • DYNAMICS
  • SADDLES

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 Henon 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.",
keywords = "STRANGE ATTRACTORS, GENERALIZED DIMENSIONS, FRACTAL MEASURES, SINGULARITIES, TRAJECTORIES, DYNAMICS, SADDLES",
author = "Lai, {Y C} and Ying-Cheng Lai",
year = "1997",
month = "12",
language = "English",
volume = "56",
pages = "6531--6539",
journal = "Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics",
issn = "1063-651X",
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TY - JOUR

T1 - Characterization of the natural measure by unstable periodic orbits in nonhyperbolic chaotic systems

AU - Lai, Y C

AU - Lai, Ying-Cheng

PY - 1997/12

Y1 - 1997/12

N2 - 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 Henon 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.

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 Henon 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.

KW - STRANGE ATTRACTORS

KW - GENERALIZED DIMENSIONS

KW - FRACTAL MEASURES

KW - SINGULARITIES

KW - TRAJECTORIES

KW - DYNAMICS

KW - SADDLES

M3 - Article

VL - 56

SP - 6531

EP - 6539

JO - Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

JF - Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

SN - 1063-651X

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ER -