The natural measure of nonattracting chaotic sets and its representation by unstable periodic orbits

Mukeshwar Dhamala, Ying-Cheng Lai

Research output: Contribution to journalArticle

7 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 the chaotic set contained in that region. This result has been shown to be valid for hyperbolic chaotic invariant sets. The aim of this paper is to examine whether this result applies to nonhyperbolic, nonattracting chaotic saddles which lead to transient chaos in physical systems. In particular, we examine, quantitatively, the closeness of the natural measure obtained from a long trajectory on the chaotic saddle to that evaluated from unstable periodic orbits embedded in the set. We also analyze the difference between the long-time average values of physical quantities evaluated with respect to a dense trajectory and those computed from unstable periodic orbits. Results with both the Henon map and the Ikeda-Hammel-Jones-Moloney 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 saddles.

Original languageEnglish
Pages (from-to)2991-3005
Number of pages15
JournalInternational Journal of Bifurcation and Chaos
Volume12
Issue number12
Publication statusPublished - Dec 2002

Keywords

  • transient chaos
  • unstable periodic orbits
  • natural measure
  • ergodic averages
  • nonhyperbolicity
  • leapfrogging vortex pairs
  • open hydrodynamical flows
  • strange attractors
  • ring cavity
  • saddles
  • dynamics
  • systems
  • dimensions
  • advection
  • trajectories

Cite this

The natural measure of nonattracting chaotic sets and its representation by unstable periodic orbits. / Dhamala, Mukeshwar; Lai, Ying-Cheng.

In: International Journal of Bifurcation and Chaos, Vol. 12, No. 12, 12.2002, p. 2991-3005.

Research output: Contribution to journalArticle

@article{955bb635c70643d3a0727e5d502c109b,
title = "The natural measure of nonattracting chaotic sets and its representation by unstable periodic orbits",
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 the chaotic set contained in that region. This result has been shown to be valid for hyperbolic chaotic invariant sets. The aim of this paper is to examine whether this result applies to nonhyperbolic, nonattracting chaotic saddles which lead to transient chaos in physical systems. In particular, we examine, quantitatively, the closeness of the natural measure obtained from a long trajectory on the chaotic saddle to that evaluated from unstable periodic orbits embedded in the set. We also analyze the difference between the long-time average values of physical quantities evaluated with respect to a dense trajectory and those computed from unstable periodic orbits. Results with both the Henon map and the Ikeda-Hammel-Jones-Moloney 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 saddles.",
keywords = "transient chaos, unstable periodic orbits, natural measure, ergodic averages, nonhyperbolicity, leapfrogging vortex pairs, open hydrodynamical flows, strange attractors, ring cavity, saddles, dynamics, systems, dimensions, advection, trajectories",
author = "Mukeshwar Dhamala and Ying-Cheng Lai",
year = "2002",
month = "12",
language = "English",
volume = "12",
pages = "2991--3005",
journal = "International Journal of Bifurcation and Chaos",
issn = "0218-1274",
publisher = "WORLD SCIENTIFIC PUBL CO PTE LTD",
number = "12",

}

TY - JOUR

T1 - The natural measure of nonattracting chaotic sets and its representation by unstable periodic orbits

AU - Dhamala, Mukeshwar

AU - Lai, Ying-Cheng

PY - 2002/12

Y1 - 2002/12

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 the chaotic set contained in that region. This result has been shown to be valid for hyperbolic chaotic invariant sets. The aim of this paper is to examine whether this result applies to nonhyperbolic, nonattracting chaotic saddles which lead to transient chaos in physical systems. In particular, we examine, quantitatively, the closeness of the natural measure obtained from a long trajectory on the chaotic saddle to that evaluated from unstable periodic orbits embedded in the set. We also analyze the difference between the long-time average values of physical quantities evaluated with respect to a dense trajectory and those computed from unstable periodic orbits. Results with both the Henon map and the Ikeda-Hammel-Jones-Moloney 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 saddles.

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 the chaotic set contained in that region. This result has been shown to be valid for hyperbolic chaotic invariant sets. The aim of this paper is to examine whether this result applies to nonhyperbolic, nonattracting chaotic saddles which lead to transient chaos in physical systems. In particular, we examine, quantitatively, the closeness of the natural measure obtained from a long trajectory on the chaotic saddle to that evaluated from unstable periodic orbits embedded in the set. We also analyze the difference between the long-time average values of physical quantities evaluated with respect to a dense trajectory and those computed from unstable periodic orbits. Results with both the Henon map and the Ikeda-Hammel-Jones-Moloney 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 saddles.

KW - transient chaos

KW - unstable periodic orbits

KW - natural measure

KW - ergodic averages

KW - nonhyperbolicity

KW - leapfrogging vortex pairs

KW - open hydrodynamical flows

KW - strange attractors

KW - ring cavity

KW - saddles

KW - dynamics

KW - systems

KW - dimensions

KW - advection

KW - trajectories

M3 - Article

VL - 12

SP - 2991

EP - 3005

JO - International Journal of Bifurcation and Chaos

JF - International Journal of Bifurcation and Chaos

SN - 0218-1274

IS - 12

ER -