Measuring the strangeness of strange attractors

Grassberger, Peter; Procaccia, Itamar

doi:10.1016/0167-2789(83)90298-1

Cited by 4,300 publications

(1,461 citation statements)

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“…The notion of correlation dimension, introduced by Grassberger and Procaccia [21,22] , suits well experimental situations, when only a single time series is available. It is now being used widely in many branches of physical science.…”

Section: Correlation Dimension and Hurst Exponentmentioning

confidence: 99%

Time series model based on global structure of complete genome

Anh

2001

Chaos, Solitons & Fractals

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Section: Correlation Dimension and Hurst Exponentmentioning

confidence: 99%

Time series model based on global structure of complete genome

Anh

2001

Chaos, Solitons & Fractals

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“…In Table 3 our computational data on the Lyapunov's exponents, Kaplan-York attractor dimensions, the Kolmogorov entropy K entr are listed. computed using the Grassberger-Procaccia algorithm [25]. The Kaplan-York dimension is less than the embedding dimension that confirms the correct choice of the latter.…”

Section: Results and Conclusionmentioning

confidence: 84%

“…The first correlation integral analysis uses the correlation integral, C(r), to distinguish between chaotic and stochastic systems. To compute the correlation integral, the algorithm of Grassberger and Procaccia [25] is the most commonly used approach, where the correlation integral is where H is the Heaviside step function with H(u) = 1 for u > 0 and H(u) = 0 for u  0, r is the radius of sphere centered on y i or y j , and N is the number of data measurements. If the time series is characterized by an attractor, then the integral C(r) is related to the radius r given by…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of laser systems with elements of a chaos: Advanced computational code

Buyadzhi

Глушков

Khetselius

et al. 2017

J. Phys.: Conf. Ser.

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Abstract. A general, uniform chaos-geometric computational approach to analysis, modelling and prediction of the non-linear dynamics of quantum and laser systems (laser and quantum generators system etc) with elements of the deterministic chaos is briefly presented. The approach is based on using the advanced generalized techniques such as the wavelet analysis, multi-fractal formalism, mutual information approach, correlation integral analysis, false nearest neighbour algorithm, the Lyapunov's exponents analysis, and surrogate data method, prediction models etc There are firstly presented the numerical data on the topological and dynamical invariants (in particular, the correlation, embedding, Kaplan-York dimensions, the Lyapunov's exponents, Kolmogorov's entropy and other parameters) for laser system (the semiconductor GaAs/GaAlAs laser with a retarded feedback) dynamics in a chaotic and hyperchaotic regimes. IntroductionIn a modern computational quantum and laser physics, electronics and others there are studied various systems and devices (such as atomic and molecular systems in an electromagnetic field, multi-element semiconductors and gas lasers etc), dynamics of which can exhibit a chaotic behaviour. These systems can be considered in the first approximation as a grid of autogenerators (quantum generators), coupled by different way [1][2][3][4][5][6][7][8][9][10]. It is easily to understand that a quantitative studying of the chaos phenomenon features is of a great interest and importance for many scientific and technical applications. At the present time it became one of the most actual and important problems of computational physics of the complex non-linear systems.In this work we firstly applied a general, uniform chaos-geometric formalism to analysis and modelling of non-linear dynamics of the laser systems with elements of a chaos. The formalism is based on using the advanced generalized techniques such as the wavelet analysis, multi-fractal formalism, mutual information approach, correlation integral analysis, false nearest neighbour algorithm, the Lyapunov's exponents analysis, and surrogate data method, prediction models etc (see details in Refs. [6][7][8][9][10][11][12][13][14][15][16][17][18][19]). There are firstly presented the numerical data on topological and dynamical invariants of chaotic systems, in particular, the correlation, embedding, Kaplan-York dimensions, the Lyapunov's exponents, Kolmogorov's entropy etc for laser (the semiconductor GaAs/GaAlAs laser with retarded feedback) systems dynamics in chaotic and hyperchaotic regimes.

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“…On the other hand, the correlation dimension is computed by the sphere-counting algorithm by Grassberger-Procaccia 14 . Briefly, it is obtained by counting the data points inside hyperspheres of various radii centered on each data point in a reconstructed phase space with some embedding dimension.…”

Section: Resultsmentioning

confidence: 99%

Chaotic oscillations in a lipid doped membrane

Delgado¹,

Medina²

2001

J. Braz. Chem. Soc.

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O potencial elétrico de uma membrana dopada com trioleína e colocada entre duas soluções aquosas de cloretos de sódio e potássio foi analisada para verificar a ocorrência de caos determinístico, usando métodos padrão de teoria de sistemas dinâmicos não lineares. Os resultados fornecem evidências sólidas de que este sistema apresenta dinâmica caótica, e não é apenas um exemplo de ruído experimental.The electric potential of a membrane doped with triolein, placed between aqueous solutions of sodium chloride and potassium chloride, was analyzed for the existence of deterministic chaos using standard methods of nonlinear dynamics system theory. The results provide solid evidence that the system shows a chaotic dynamics and not just an example of experimental noise in the data.

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Measuring the strangeness of strange attractors

Cited by 4,300 publications

References 20 publications

Time series model based on global structure of complete genome

Time series model based on global structure of complete genome

Nonlinear dynamics of laser systems with elements of a chaos: Advanced computational code

Chaotic oscillations in a lipid doped membrane

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