### 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