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.

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.

CHAOTIC ANALYSIS OF A TIME SERIES COMPOSED OF SEISMIC EVENTS RECORDED IN JAPAN

1994 ◽  
Vol 04 (01) ◽  
pp. 87-98 ◽  
Author(s):  
G.P. PAVLOS ◽  
L. KARAKATSANIS ◽  
J.B. LATOUSSAKIS ◽  
D. DIALETIS ◽  
G. PAPAIOANNOU
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Initial Conditions ◽  
Low Frequency ◽  
Underlying Mechanism ◽  
Largest Lyapunov Exponent ◽  
Seismic Events ◽  
Earthquake Dynamics ◽  
The Largest Lyapunov Exponent ◽  
Low Dimensional

A chaotic analysis approach was applied to an earthquake time series recorded in the Japanese area in order to test the assumption that the earthquake process could be the manifestation of a chaotic low dimensional process. For the study of the seismicity we have used a time series consisting of time differences between two consecutive seismic events with magnitudes greater than 2.6. The results of our study show that the underlying mechanism, as expressed by the time series, can be described by low dimensional chaotic dynamics. The power spectrum of the time series shows a power law profile with two slopes, α=1.4 in the low frequency and α=0.05 in the high frequency regions, while the slopes of the correlation integrals show an apparent plateau at the scaling region, which saturates at the value D≈3.2. The largest Lyapunov exponent was found to be ≈0.9. The positive value of the largest Lyapunov exponent reveals strong sensitivity to initial conditions of the supposed earthquake dynamics.

scholarly journals The Dynamics of Philippine Foreign Exchange Rates (2013-2017): A Test for Chaos

2019 ◽  
Vol 8 (11) ◽  
pp. 486-489
Keyword(s):  
Time Series ◽  
Exchange Rate ◽  
Lyapunov Exponent ◽  
Exchange Rates ◽  
Foreign Exchange ◽  
Chaotic Behavior ◽  
The Philippines ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
Foreign Exchange Rates

In most studies on dynamics of time series financial data, the absence of chaotic behavior is generally observed. However, this theory is not yet established in the dynamics of foreign exchange rates. Conflicting claims of presence and absence of chaos in foreign exchange rates open door for further investigation considering various deterministic factors. This work examines the dynamics of exchange rate of the Philippine Peso against selected foreign currencies. Time series data were collected for eight (8) of Philippine’s top trading partners as categorized according to economic condition. The data obtained with permission from the Central Bank of the Philippines covered the years 2013 to 2017. Data sets were plotted revealing non-linear movement of Philippine exchange rates against time. The foreign exchange rate time series obtained per currency were examined for chaotic behavior by computing the Largest Lyapunov Exponents (LLE). A positive Lyapunov exponent is an indication of sensitivity dependence, i.e, a chaotic dynamics; whereas, a negative Lyapunov exponent indicates otherwise. Computed LLE’s varied per currency but all were found to be negative. Therefore, using the Largest Lyapunov Exponent Test (LLE), analysis of the time series of Philippine foreign exchange rates shows little evidence of chaotic patterns.

scholarly journals A Robust Method to Estimate the Largest Lyapunov Exponent of Noisy Signals: A Revision to the Rosenstein’s Algorithm

2018 ◽  
Author(s):  
Sina Mehdizadeh
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Human Walking ◽  
Largest Lyapunov Exponent ◽  
Robust Method ◽  
Noisy Signals ◽  
Experimental Time Series ◽  
Data Points ◽  
Robust To Noise ◽  
Walking Time

AbstractAimThis study proposed a revision to the Rosenstein’s method of numerical calculation of largest Lyapunov exponent (LyE) to make it more robust to noise.MethodsTo this aim, the effect of increasing number of initial neighboring points on the LyE value was investigated and compared to the values obtained by filtering the time series. Both simulated (Lorenz and passive dynamic walker) and experimental (human walking) time series were used to calculate LyE. The number of initial neighbors used to calculate LyE for all time series was 1 (the original Rosenstein’s method), 2, 3, 4, 5, 10, 15, 20, 25, and 30 data points.ResultsThe results demonstrated that the LyE graph reached a plateau at the 15-point neighboring condition inferring that the LyE values calculated using at least 15 neighboring points were consistent and reliable.ConclusionThe proposed method could be used to calculate LyE more reliably in experimental time series acquired from biological systems where noise is omnipresent.

The Dynamics of a Flexible Levitated Droplet

2001 ◽  
Author(s):  
Mihai Dupac ◽  
Dan B. Marghitu ◽  
David G. Beale
Keyword(s):  
Nonlinear Dynamics ◽  
Time Series ◽  
Lyapunov Exponent ◽  
Time Series Data ◽  
Simulated Data ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
Dynamics Analysis ◽  
Attractor Dimension ◽  
Levitated Droplet

Abstract In this paper, a nonlinear dynamics analysis of the simulated data was considered to study the time evolution of an electro-magnetically levitated flexible droplet. The main goals of this work are to study the behavior of the levitated droplet and to investigate its stability. Quantities characterizing time series data such as attractor dimension or largest Lyapunov exponent were computed.

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