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.

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.

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.

Study of Rehabilitation of Injured Knee Joint Applying Chaotic Theory in Human Body Motion

2007 ◽  
Vol 342-343 ◽  
pp. 581-584
Author(s):  
Byung Young Moon ◽  
Kwon Son ◽  
Jung Hong Park
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Angular Displacement ◽  
Three Dimensional ◽  
Gait Pattern ◽  
Body Motion ◽  
Largest Lyapunov Exponent ◽  
Chaos Analysis ◽  
Nonlinear Technique ◽  
Injured Knee

Gait analysis is essential to identify accurate cause and knee condition from patients who display abnormal walking. Traditional linear tools can, however, mask the true structure of motor variability, since biomechanical data from a few strides during the gait have limitation to understanding the system. Therefore, it is necessary to propose a more precise dynamic method. The chaos analysis, a nonlinear technique, focuses on understanding how variations in the gait pattern change over time. Healthy eight subjects walked on a treadmill for 100 seconds at 60 Hz. Three dimensional walking kinematic data were obtained using two cameras and KWON3D motion analyzer. The largest Lyapunov exponent from the measured knee angular displacement time series was calculated to quantify local stability. This study quantified the variability present in time series generated from gait parameter via chaos analysis. Gait pattern is found to be chaotic. The proposed Lyapunov exponent can be used in rehabilitation and diagnosis of recoverable patients.

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.

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

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.

scholarly journals Chaos Generated by Switching Fractional Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Wang Xingyuan ◽  
Qin Xue
Keyword(s):  
Lyapunov Exponent ◽  
Future Application ◽  
Largest Lyapunov Exponent ◽  
Fractional Systems ◽  
The Future ◽  
Fractional System ◽  
First Time

We, for the first time, investigate the basic behaviours of a chaotic switching fractional system via both theoretical and numerical ways. To deeply understand the mechanism of the chaos generation, we also analyse the parameterization of the switching fractional system and the dynamics of the system's trajectory. Then we try to write down some detailed rules for designing chaotic or chaos-like systems by switching fractional systems, which can be used in the future application. At last, for the first time, we proposed a new switching fractional system, which can generate three attractors with the positive largest Lyapunov exponent.

Comprehensive Method of Chaotic Vibration Identification

2005 ◽  
Author(s):  
Shu-Yong Liu ◽  
Xiang Yu ◽  
Shi-Jian Zhu
Keyword(s):  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Time History ◽  
Chaotic Signal ◽  
Measured Signal ◽  
Chaotic State ◽  
Chaotic Vibration ◽  
Comprehensive Method ◽  
The Difference ◽  
Topologic Structure

When the parameters are in a special range, the response of a nonlinear vibration system is chaotic, which is different from the classical regular response such as primary, super-, sub-, ultra-sub-harmonic resonances. Because the chaotic time series looks like random signal, the characteristics of the chaos cannot be identified from the time history. This paper presents a comprehensive method to identify the chaotic vibration. The attractor of the system is reconstructed in the phase space, and thus the characteristics of the chaotic signal are reflected by the attractor. If the attractor is regular, the response may be periodic. If the topologic structure of the phase diagram is very complicate, the attractor is strange, that is, the system may be in a chaotic state. Correlation dimension and Lyapunov exponent are calculated to prove the conclusions above. It is clear that if the correlation dimension is fraction and the Lyapunov exponent is positive, the measured signal is chaotic. The difference between chaotic signal and noise is studied as well Results show that the comprehensive method can be applied to identify the chaotic vibration efficiently.

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