### Abstract

We address the calculation of correlation dimension, the estimation of Lyapunov exponents, and the detection of unstable periodic orbits. from transient chaotic time series. Theoretical arguments and numerical experiments show that the Grassberger-Procaccia algorithm can be used to estimate the dimension of an underlying chaotic saddle from an ensemble of chaotic transients. We also demonstrate that Lyapunov exponents can be estimated by computing the rates of separation of neighboring phase-space states constructed from each transient time series in an ensemble. Numerical experiments utilizing the statistics of recurrence times demonstrate that unstable periodic orbits of low periods can be extracted even when noise is present. In addition, we test the scaling law for the probability of finding periodic orbits. The scaling law implies that unstable periodic orbits of high period are unlikely to be detected from transient chaotic time series.

Original language | English |
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Article number | 056207 |

Number of pages | 9 |

Journal | Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 64 |

Issue number | 5 |

DOIs | |

Publication status | Published - Nov 2001 |

### Keywords

- unstable periodic-orbits
- strange attractors
- correlation dimension
- Lyapunov exponents
- power systems
- ring cavity
- saddles
- flows
- noise
- boundaries

### Cite this

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

*64*(5), [056207]. https://doi.org/10.1103/PhysRevE.64.056207

**Analyses of transient chaotic time series.** / Dhamala, Mukeshwar; Lai, Ying-Cheng; Kostelich, Eric J .

Research output: Contribution to journal › Article

*Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 64, no. 5, 056207. https://doi.org/10.1103/PhysRevE.64.056207

}

TY - JOUR

T1 - Analyses of transient chaotic time series

AU - Dhamala, Mukeshwar

AU - Lai, Ying-Cheng

AU - Kostelich, Eric J

PY - 2001/11

Y1 - 2001/11

N2 - We address the calculation of correlation dimension, the estimation of Lyapunov exponents, and the detection of unstable periodic orbits. from transient chaotic time series. Theoretical arguments and numerical experiments show that the Grassberger-Procaccia algorithm can be used to estimate the dimension of an underlying chaotic saddle from an ensemble of chaotic transients. We also demonstrate that Lyapunov exponents can be estimated by computing the rates of separation of neighboring phase-space states constructed from each transient time series in an ensemble. Numerical experiments utilizing the statistics of recurrence times demonstrate that unstable periodic orbits of low periods can be extracted even when noise is present. In addition, we test the scaling law for the probability of finding periodic orbits. The scaling law implies that unstable periodic orbits of high period are unlikely to be detected from transient chaotic time series.

AB - We address the calculation of correlation dimension, the estimation of Lyapunov exponents, and the detection of unstable periodic orbits. from transient chaotic time series. Theoretical arguments and numerical experiments show that the Grassberger-Procaccia algorithm can be used to estimate the dimension of an underlying chaotic saddle from an ensemble of chaotic transients. We also demonstrate that Lyapunov exponents can be estimated by computing the rates of separation of neighboring phase-space states constructed from each transient time series in an ensemble. Numerical experiments utilizing the statistics of recurrence times demonstrate that unstable periodic orbits of low periods can be extracted even when noise is present. In addition, we test the scaling law for the probability of finding periodic orbits. The scaling law implies that unstable periodic orbits of high period are unlikely to be detected from transient chaotic time series.

KW - unstable periodic-orbits

KW - strange attractors

KW - correlation dimension

KW - Lyapunov exponents

KW - power systems

KW - ring cavity

KW - saddles

KW - flows

KW - noise

KW - boundaries

U2 - 10.1103/PhysRevE.64.056207

DO - 10.1103/PhysRevE.64.056207

M3 - Article

VL - 64

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

IS - 5

M1 - 056207

ER -