### 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 language | English |
---|---|

Pages (from-to) | 2991-3005 |

Number of pages | 15 |

Journal | International Journal of Bifurcation and Chaos |

Volume | 12 |

Issue number | 12 |

Publication status | Published - 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

*International Journal of Bifurcation and Chaos*,

*12*(12), 2991-3005.

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

Research output: Contribution to journal › Article

*International Journal of Bifurcation and Chaos*, vol. 12, no. 12, pp. 2991-3005.

}

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 -