### Abstract

In this paper, two issues are addressed: (1) the applicability of the delay-coordinate embedding method to transient chaotic time series analysis, and (2) the Hilbert transform methodology for chaotic signal processing.

A common practice in chaotic time series analysis has been to reconstruct the phase space by utilizing the delay-coordinate embedding technique, and then to compute dynamical invariant quantities of interest such as unstable periodic orbits, the fractal dimension of the underlying chaotic set, and its Lyapunov spectrum. As a large body of literature exists on applying the technique to time series from chaotic attractors, a relatively unexplored issue is its applicability to dynamical systems that exhibit transient chaos. Our focus will be on the analysis of transient chaotic time series. We will argue and provide numerical support that the current delay-coordinate embedding techniques for extracting unstable periodic orbits, for estimating the fractal dimension, and for computing the Lyapunov exponents can be readily adapted to transient chaotic time series.

A technique that is gaining an increasing attention is the Hilbert transform method for signal processing in nonlinear systems. The general goal of the Hilbert method is to assess the spectrum of the instantaneous frequency associated with the underlying dynamical process. To obtain physically meaningful results, it is necessary for the signal to possess a proper rotational structure in the complex plane of the analytic signal constructed by the original signal and its Hilbert transform. We will describe a recent decomposition procedure for this task and apply the technique to chaotic signals. We will also provide an example to demonstrate that the methodology can be useful for addressing some fundamental problems in chaotic dynamics.

Original language | English |
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Pages (from-to) | 1383-1422 |

Number of pages | 40 |

Journal | International Journal of Bifurcation and Chaos |

Volume | 13 |

Issue number | 6 |

Publication status | Published - Jun 2003 |

### Keywords

- chaotic time series
- delay coordinates
- embedding
- transient chaos
- correlation dimension
- Lyapunov exponents
- unstable periodic orbits
- Hiebert transform
- instantaneous frequency
- unstable periodic-orbits
- open hydrodynamical flows
- fractal basin boundaries
- low-dimensional dynamics
- strange attractors
- phase synchronization
- natural measure
- delay time

## Cite this

*International Journal of Bifurcation and Chaos*,

*13*(6), 1383-1422.