A program for the user-independent computation of the correlation dimension and the largest Lyapunov exponent of heart rate dynamics from small data sets

1996 ◽  
Vol 49 (1) ◽  
pp. 61-68 ◽  
Author(s):  
R.E. Ganz ◽  
C. Lenz
Keyword(s):  
Heart Rate ◽  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Largest Lyapunov Exponent ◽  
Small Data ◽  
Data Sets ◽  
Heart Rate Dynamics ◽  
Small Data Sets ◽  
The Largest Lyapunov Exponent

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.

Delay time window and plateau onset of the correlation dimension for small data sets

1998 ◽  
Vol 58 (5) ◽  
pp. 5676-5682 ◽  
Author(s):  
H. S. Kim ◽  
R. Eykholt ◽  
J. D. Salas
Keyword(s):  
Delay Time ◽  
Correlation Dimension ◽  
Time Window ◽  
Small Data ◽  
Data Sets ◽  
Small Data Sets

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.

Dynamic Prediction of Rolling Bearing Friction Torque Using Lyapunov Exponent Method

2010 ◽  
Vol 44-47 ◽  
pp. 1120-1124 ◽  
Author(s):  
Xin Tao Xia ◽  
Tao Mei Lv
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Rolling Bearing ◽  
Friction Torque ◽  
Small Data ◽  
Data Sets ◽  
Dynamic Prediction ◽  
Information Method ◽  
Small Data Sets ◽  
Bearing Friction

Based on the chaos theory, the Lyapunov exponent method is employed to predict dynamically the time series of the rolling bearing friction torque. First, the embedding dimension and the delay time for the phase space reconstruction are estimated with the Cao method and the mutual information method, respectively. Second, the maximum of the Lyapunov exponents is calculated by small data sets. Lastly, the nearest neighboring point is sought via the Euclidean distance. The experimental investigation shows that the method proposed in this paper is able to forecast effectively the rolling bearing friction torque as a time series, only with very small predicted error.

scholarly journals VALIDATING IDENTIFIED NONLINEAR MODELS WITH CHAOTIC DYNAMICS

1994 ◽  
Vol 04 (01) ◽  
pp. 109-125 ◽  
Author(s):  
LUIS A. AGUIRRE ◽  
S.A. BILLINGS
Keyword(s):  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Chaotic Dynamics ◽  
Nonlinear Models ◽  
Original System ◽  
Largest Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Poincaré Sections ◽  
The Largest Lyapunov Exponent ◽  
Poincare Sections

This paper investigates the effectiveness of several criteria for validating models which exhibit chaotic dynamics. Embedded trajectories, Poincaré sections, bifurcation diagrams, the largest Lyapunov exponent and correlation dimension are considered. The Duffing-Ueda equation and four identified models are used as examples. The results show that models with similar invariants such as Poincaré sections, the largest Lyapunov exponent and correlation dimension may have very different bifurcation behaviours. This suggests that the requirement that an identified model should reproduce the bifurcation pattern of the original system is a very exacting criterion which is well suited for validation purposes.

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.

The Lyapunov Exponent Variance of an Electronic Manufacturing Enterprise’s Daily Qualified Rate Time Series by Improved Small Data Sets Approach

2012 ◽  
Vol 197 ◽  
pp. 271-277
Author(s):  
Zhu Ping Gong
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Chaotic Time Series ◽  
Quality System ◽  
Quality Level ◽  
Largest Lyapunov Exponent ◽  
Small Data ◽  
Data Set ◽  
Electronic Manufacturing ◽  
Small Data Set

Small data set approach is used for the estimation of Largest Lyapunov Exponent (LLE). Primarily, the mean period drawback of Small data set was corrected. On this base, the LLEs of daily qualified rate time series of HZ, an electronic manufacturing enterprise, were estimated and all positive LLEs were taken which indicate that this time series is a chaotic time series and the corresponding produce process is a chaotic process. The variance of the LLEs revealed the struggle between the divergence nature of quality system and quality control effort. LLEs showed sharp increase in getting worse quality level coincide with the company shutdown. HZ’s daily qualified rate, a chaotic time series, shows us the predictable nature of quality system in a short-run.

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