The Takens Embedding Theorem

Noakes, Lyle

doi:10.1142/s0218127491000634

Cited by 102 publications

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“…This idea might be motivated by the usage of displacement and velocity as states in oscillators. Takens has proven that this method will produce embeddings, provided that the system is sufficiently smooth and noise free (see also Noakes [5]). However, real systems are not noise-free.…”

Section: Introductionmentioning

confidence: 99%

Fractional Derivatives Applied to Phase-Space Reconstructions

Feeny

Lin

2004

Nonlinear Dyn

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The concept and application of phase-space reconstructions are reviewed. Fractional derivatives are then proposed for the purpose of reconstructing dynamics from a single observed time history. A procedure is presented in which the fractional derivatives of time series data are obtained in the frequency domain. The method is applied to the Lorenz system. The ability of the method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and correlation dimensions. The reconstruction is compared to the commonly used method of delays in order to assess the choice of reconstruction parameters, and also the quality of results.

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Section: Introductionmentioning

confidence: 99%

Fractional Derivatives Applied to Phase-Space Reconstructions

Feeny

Lin

2004

Nonlinear Dyn

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“…To realize dynamic characteristic extraction of the object time series [23], [24], we will estimate the behaviors of the hidden deterministic state variables Xi (i = 1, 2) of the actual observation time series. To estimate the deterministic rule, the reconstruction theorem by F. Takens [25], [26] is used. 6 https://www.worldscientific.com/doi/abs/10.1142/S0218127491000634 7 https://en.wikipedia.org/wiki/Takens%27s_theorem 8 https://en.wikipedia.org/wiki/El_Niño-Southern_Oscillation 9 http://www.sosmath.com/calculus/diff/der07/der07.html …”

Section: Proposed Methodsmentioning

confidence: 99%

“…Meanwhile, detecting strange attractors in turbulence in Dynamical Systems and Turbulence [25] is discussed. The Takens embedding theorem 6 is introduced [26].…”

Section: Introductionmentioning

confidence: 99%

El Niño / La Niña Identification based on Takens Reconstruction Theory

Arai¹,

Seto²

2018

ijacsa

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Abstract-An identification method for earth observation data according to a chaotic behavior based on Takens reconstruction theory is proposed. The proposed method is examined by using the observed time series data of SST (Sea Surface Temperature) and the SOI (Southern Oscillation Index) data. The experimental results show that the time for the identification of the proposed method is not later than that of the existing method. Author confirmed that by using the definitions of the Japan Meteorological Agency and the use of Equations, I can identify El Niño / La Niña at an earlier time. In other words, we do not necessarily need a numerical value for 10 months by identifying the proposed method. I confirmed that the time required for the identification judgment of the proposed method is about one month. The proposed method is not based on extrapolation method with numerical model or governing equation, but based on interpolation method using only actual observation time series.

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“…) > represents the time-shifted observations of the dynamics induced by M on M. An in-depth and detailed discussion of this subjected can be found in Deyle andSugihara (2011) andNoakes (1991).…”

Section: The Wavelet Phase-space Reconstruction and The Scale Dependementioning

confidence: 99%

Time Series Chaos Detection and Assessment via Scale Dependent Lyapunov Exponent

Fenga¹

2016

IJSP

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Many dynamical systems in a wide range of disciplines -such as engineering, economy and biology -exhibit complex behaviors generated by nonlinear components which might result in deterministic chaos. While in lab-controlled setups its detection and level estimation is in general a doable task, usually the same does not hold for many practical applications. This is because experimental conditions imply facts like low signal-to-noise ratios, small sample sizes and not-repeatability of the experiment, so that the performances of the tools commonly employed for chaos detection can be seriously affected. To tackle this problem, a combined approach based on wavelet and chaos theory is proposed. This is a procedure designed to provide the analyst with qualitative and quantitative information, hopefully conducive to a better understanding of the dynamical system the time series under investigation is generated from. The chaos detector considered is the well known Lyapunov Exponent. A real life application, using the Italian Electric Market price index, is employed to corroborate the validity of the proposed approach.

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The Takens Embedding Theorem

Cited by 102 publications

References 0 publications

Fractional Derivatives Applied to Phase-Space Reconstructions

Fractional Derivatives Applied to Phase-Space Reconstructions

El Niño / La Niña Identification based on Takens Reconstruction Theory

Time Series Chaos Detection and Assessment via Scale Dependent Lyapunov Exponent

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