scholarly journals Correlation Dimension and Largest Lyapunov Exponent for Broadband Edge Turbulence in the Compact Helical System

1994 ◽  
Vol 73 (5) ◽  
pp. 660-663 ◽  
Author(s):  
A. Komori ◽  
T. Baba ◽  
T. Morisaki ◽  
M. Kono ◽  
H. Iguchi ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Largest Lyapunov Exponent

EFFECTS OF TREND AND PERIODICITY ON THE CORRELATION DIMENSION AND THE LYAPUNOV EXPONENTS

2008 ◽  
Vol 18 (12) ◽  
pp. 3679-3687 ◽  
Author(s):  
AYDIN A. CECEN ◽  
CAHIT ERKAL
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Time Series Data ◽  
Logistic Map ◽  
Model Identification ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
The Impact

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

Research of Extracting Information about Reservoir Logging Signals Based on Chaos Characteristics

2013 ◽  
Vol 380-384 ◽  
pp. 3742-3745
Author(s):  
Chun Yan Nie ◽  
Rui Li ◽  
Wan Li Zhang
Keyword(s):  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Approximate Entropy ◽  
Water Layer ◽  
Time Correlation ◽  
Largest Lyapunov Exponent ◽  
Characteristic Parameters ◽  
Chaotic Characteristic ◽  
Oil Water

The mechanism of logging signals generating was researched. In the same time, correlation dimension, largest Lyapunov exponent and approximate entropy of chaotic characteristics were extracted. On this basis, chaotic characteristic parameters were applied in processing, analysis and interpretation, try to find chaotic characteristics of different of reservoirs for example oil, water layer and the dry layer. The results showed that chaos characteristics in different reservoir is different, therefore, we can distinguish the different natures of reservoirs by extracting chaos characteristics.

Chaotic State of Physical Pendulum Driven by Anharmonic Periodic Force

2003 ◽  
Vol 17 (22n24) ◽  
pp. 4321-4326
Author(s):  
Xiao-Ping Qin ◽  
Zheng-Mao Sheng
Keyword(s):  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Largest Lyapunov Exponent ◽  
Physical Pendulum ◽  
Chaotic State ◽  
Periodic Force ◽  
The Largest Lyapunov Exponent ◽  
Experiment And Simulation

The chaotic movement of physical pendulum, which is driven by an anharmonic periodic force, is studied by experiment and simulation. The correlation dimension and the largest Lyapunov exponent is obtained by numerical simulation.It is found that there is an obvious difference of correlation dimensions between the systems driven by anharmonic periodic force and harmonic periodic force.

Analog Neural Network Model Produces Chaos Similar to the Human EEG

1997 ◽  
Vol 07 (05) ◽  
pp. 1133-1140 ◽  
Author(s):  
Vladimir E. Bondarenko
Keyword(s):  
Neural Network ◽  
Lyapunov Exponent ◽  
Network Model ◽  
Shannon Entropy ◽  
Neural Network Model ◽  
Correlation Dimension ◽  
The Self ◽  
Self Organization ◽  
Largest Lyapunov Exponent ◽  
Human Eeg

The self-organization processes in an analog asymmetric neural network with the time delay were considered. It was shown that in dependence on the value of coupling constants between neurons the neural network produced sinusoidal, quasi-periodic or chaotic outputs. The correlation dimension, largest Lyapunov exponent, Shannon entropy and normalized Shannon entropy of the solutions were studied from the point of view of the self-organization processes in systems far from equilibrium state. The quantitative characteristics of the chaotic outputs were compared with the human EEG characteristics. The calculation of the correlation dimension ν shows that its value is varied from 1.0 in case of sinusoidal oscillations to 9.5 in chaotic case. These values of ν agree with the experimental values from 6 to 8 obtained from the human EEG. The largest Lyapunov exponent λ calculated from neural network model is in the range from -0.2 s -1 to 4.8 s -1 for the chaotic solutions. It is also in the interval from 0.028 s -1 to 2.9 s -1 of λ which is observed in experimental study of the human EEG.

CHAOTIC INDICATORS AND GAUSSIAN RANDOM PROCESSES: SOME SURPRISING RESULTS

Fractals ◽  
1996 ◽  
Vol 04 (01) ◽  
pp. 73-90 ◽  
Author(s):  
L. BERGAMASCO ◽  
M. SERIO
Keyword(s):  
Lyapunov Exponent ◽  
Surface Waves ◽  
Gaussian Processes ◽  
Correlation Dimension ◽  
Power Spectra ◽  
Ocean Surface ◽  
Largest Lyapunov Exponent ◽  
Ocean Surface Waves ◽  
The Largest Lyapunov Exponent ◽  
Parameter Ranges

The search for low-dimensional chaos in ocean surface waves is nowadays a very active field. The interpretation of the results, however, is not always straightforward. The issue addressed in this paper is how time series analysis tools from dynamical systems theory behave for a class of Gaussian processes often used in the study of ocean surface waves. The study includes the largest Lyapunov exponent, the Grassberger and Procaccia correlation dimension and the self-similarity properties. Surprisingly, for certain parameter ranges, the correlation dimension is found to be finite, the largest Lyapunov exponent is found to be positive and structure appears on all time scales. These results suggest that improved techniques and data analysis procedures may be required in order to study chaos properties of ocean surface waves or of other Gaussian processes with similar power spectra.

On the Correlation Dimension of Discrete Fractional Chaotic Systems

2020 ◽  
Vol 30 (12) ◽  
pp. 2050174 ◽  
Author(s):  
Li Ma ◽  
Xianggang Liu ◽  
Xiaotong Liu ◽  
Ying Zhang ◽  
Yu Qiu ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Chaotic Systems ◽  
Jacobian Matrix ◽  
Three Dimensional ◽  
Largest Lyapunov Exponent ◽  
Embedding Dimension ◽  
Chaotic State ◽  
Fractional Systems ◽  
With Memory

This paper is mainly devoted to the investigation of discrete-time fractional systems in three aspects. Firstly, the fractional Bogdanov map with memory effect in Riemann–Liouville sense is obtained. Then, via constructing suitable controllers, the fractional Bogdanov map is shown to undergo a transition from regular state to chaotic one. Meanwhile, the positive largest Lyapunov exponent is calculated by the Jacobian matrix algorithm to distinguish the chaotic areas. Finally, the Grassberger–Procaccia algorithm is employed to evaluate the correlation dimension of the controlled fractional Bogdanov system under different parameters. The main results show that the correlation dimension converges to a fixed value as the embedding dimension increases for the controlled fractional Bogdanov map in chaotic state, which also coincides with the conclusion driven by the largest Lyapunov exponent. Moreover, three-dimensional fractional Stefanski map is considered to further verify the effectiveness and generality of the obtained results.

The Chaotic Attractor of the Sediment Movement

Fractals ◽  
1998 ◽  
Vol 06 (02) ◽  
pp. 191-196
Author(s):  
Fengsu Chen ◽  
Kongxian Xue ◽  
Wenkang Cai
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Chaotic Attractor ◽  
Yangtze River ◽  
Correlation Dimension ◽  
Chaotic Behavior ◽  
Largest Lyapunov Exponent ◽  
The Yangtze River ◽  
Sediment Movement ◽  
Reconstructed Phase Space

We consider the chaotic behavior of the sediment movement with the observed data of the Yangtze River in China and the method of the reconstructed phase space and we find that in the sediment movement there is an attractor. As far as the real example mentioned in this paper is concerned, the correlation dimension and the largest Lyapunov exponent are around 6.6 and 0.013 respectively. These results are crucially referential for estimating the mode of the sediment movement, designing the scheme of the sediment observation, and studying the predictability problem of the sediment.

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