Analyses of transient chaotic time series

Mukeshwar Dhamala, Ying-Cheng Lai, Eric Kostelich

Research output: Contribution to journalArticle

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 languageEnglish (US)
Number of pages1
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume64
Issue number5
DOIs
StatePublished - Jan 1 2001

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Chaotic Time Series
Periodic Orbits
orbits
Unstable
Scaling Laws
Lyapunov Exponent
scaling laws
Ensemble
Numerical Experiment
exponents
Correlation Dimension
saddles
Saddle
Recurrence
Demonstrate
Phase Space
Time series
statistics
Statistics
Imply

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics

Cite this

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title = "Analyses of transient chaotic time series",
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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.

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