Analytic signals and the transition to chaos in deterministic flows

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18 Citations (Scopus)

Abstract

The transition from regular to chaotic motions in deterministic flows is characterized by a change from a discrete Fourier spectrum to a broadband one. The onset of chaos is thus associated with the creation of an infinite number of new Fourier modes. Given a system that generates a time series x(t), we study the transition to chaos from the perspective of analytic signals, which are defined via the Hilbert transform. In order to identify distinct analytic signals, we decompose the original time series x(t) into a finite number of modes that correspond to proper rotations in the complex plane of their analytic signals. We provide numerical evidence that at the transition, there is no substantial change in the number of analytic signals characterizing x(t). Furthermore, the distributions of the instantaneous frequencies of the analytic signals in the chaotic regime are well localized and exhibit no broadband feature. These results suggest a simple organization of chaos in terms of analytic signals.

Original languageEnglish (US)
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume58
Issue number6 SUPPL. A
StatePublished - 1998
Externally publishedYes

Fingerprint

Analytic Signal
chaos
Chaos
Broadband
Time series
broadband
Instantaneous Frequency
Fourier Spectrum
Hilbert Transform
Chaotic Motion
Discrete Spectrum
Argand diagram
Distinct
Decompose

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Condensed Matter Physics
  • Statistical and Nonlinear Physics

Cite this

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abstract = "The transition from regular to chaotic motions in deterministic flows is characterized by a change from a discrete Fourier spectrum to a broadband one. The onset of chaos is thus associated with the creation of an infinite number of new Fourier modes. Given a system that generates a time series x(t), we study the transition to chaos from the perspective of analytic signals, which are defined via the Hilbert transform. In order to identify distinct analytic signals, we decompose the original time series x(t) into a finite number of modes that correspond to proper rotations in the complex plane of their analytic signals. We provide numerical evidence that at the transition, there is no substantial change in the number of analytic signals characterizing x(t). Furthermore, the distributions of the instantaneous frequencies of the analytic signals in the chaotic regime are well localized and exhibit no broadband feature. These results suggest a simple organization of chaos in terms of analytic signals.",
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AB - The transition from regular to chaotic motions in deterministic flows is characterized by a change from a discrete Fourier spectrum to a broadband one. The onset of chaos is thus associated with the creation of an infinite number of new Fourier modes. Given a system that generates a time series x(t), we study the transition to chaos from the perspective of analytic signals, which are defined via the Hilbert transform. In order to identify distinct analytic signals, we decompose the original time series x(t) into a finite number of modes that correspond to proper rotations in the complex plane of their analytic signals. We provide numerical evidence that at the transition, there is no substantial change in the number of analytic signals characterizing x(t). Furthermore, the distributions of the instantaneous frequencies of the analytic signals in the chaotic regime are well localized and exhibit no broadband feature. These results suggest a simple organization of chaos in terms of analytic signals.

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