Study of Non-Smooth Gear Based on MÜLLER Algorithm

2014 ◽  
Vol 912-914 ◽  
pp. 769-773
Author(s):  
Miao Xin Xiao ◽  
Jian Jun Yang
Keyword(s):  
Lyapunov Exponent ◽  
Power Systems ◽  
Smooth Point ◽  
Gear System ◽  
Largest Lyapunov Exponent ◽  
Maximal Lyapunov Exponent ◽  
Detection Function ◽  
Unified Algorithm ◽  
The Relationship ◽  
System Phase

It is found that the relationship the detection function and collision function of non smooth point by using MÜLLER algorithm, and launched the conversion between the different states of gear system of the offset vector. So the unified algorithm of maximal Lyapunov exponent of non smooth gear system is obtained. An example is given to verified this method. The validities of algorithm for largest Lyapunov exponent of gear power systems is proved by comparing the the largest Lyapunov, system phase and Poincaré section.

scholarly journals Temporal Change of Spectra and Lyapunov Exponent Volcanic Tremor at Raung Volcano, Indonesia

2020 ◽  
Vol 9 (3S) ◽  
pp. 163-167
Keyword(s):  
Lyapunov Exponent ◽  
Fourier Transformation ◽  
Temporal Change ◽  
Volcanic Tremor ◽  
Fast Fourier Transformation ◽  
Maximal Lyapunov Exponent ◽  
Stretching Factor ◽  
Deterministic Processes ◽  
The Relationship ◽  
Generating System

Spectral Analyses and estimation maximal lyapunov exponent (MLE) of volcanic tremor recorded at Raung Volcano were carried out to investigate dynamical systems regarding to their generating system. Their results of both analyses can explain the temporal change in frequency and deterministic processes of the dynamical system. Spectral analysis of volcanic tremor was estimated by the average periodogram method which includes division, Fast Fourier Transformation and averaging. MLE was estimated by graphing the relationship between Stretching Factor (S) and the number of points in the tractor (N) diagram. Content frequency of volcanic tremor Raung Volcano is range from 2.68 to 3.7 Hz. Temporally, there is no significant change, which means that there is no change in the geometry of the Raung volcanic tremor source. This is also shown by the maximal lyapunov exponent which is temporarily constant and positive. That shows that the source process of Raung volcano is chaotic

scholarly journals Analysis of Human Standing Balance by Largest Lyapunov Exponent

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Kun Liu ◽  
Hongrui Wang ◽  
Jinzhuang Xiao ◽  
Zahari Taha
Keyword(s):  
Lyapunov Exponent ◽  
Standing Balance ◽  
Largest Lyapunov Exponent ◽  
Body Segments ◽  
Sinusoidal Stimulus ◽  
Sinusoidal Motion ◽  
Balance Ability ◽  
Human Balance ◽  
The Largest Lyapunov Exponent ◽  
The Relationship

The purpose of this research is to analyse the relationship between nonlinear dynamic character and individuals’ standing balance by the largest Lyapunov exponent, which is regarded as a metric for assessing standing balance. According to previous study, the largest Lyapunov exponent from centre of pressure time series could not well quantify the human balance ability. In this research, two improvements were made. Firstly, an external stimulus was applied to feet in the form of continuous horizontal sinusoidal motion by a moving platform. Secondly, a multiaccelerometer subsystem was adopted. Twenty healthy volunteers participated in this experiment. A new metric, coordinated largest Lyapunov exponent was proposed, which reflected the relationship of body segments by integrating multidimensional largest Lyapunov exponent values. By using this metric in actual standing performance under sinusoidal stimulus, an obvious relationship between the new metric and the actual balance ability was found in the majority of the subjects. These results show that the sinusoidal stimulus can make human balance characteristics more obvious, which is beneficial to assess balance, and balance is determined by the ability of coordinating all body segments.

Chaotic Behavior of an Inelastic Beam

1998 ◽  
Vol 14 (4) ◽  
pp. 217-221
Author(s):  
Jiin-Po Yeh
Keyword(s):  
Lyapunov Exponent ◽  
Autocorrelation Function ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Time History ◽  
Harmonic Excitation ◽  
Largest Lyapunov Exponent ◽  
Bifurcation Phenomenon ◽  
Data Points ◽  
The Relationship

ABSTRACTThe dynamical system considered in this paper is an inelastic beam whose supports are subjected to a harmonic excitation. This paper first explores whether the system has chaotic motion. The appearance of the irregular time history, strange attractor on the Poincaré map as well as period-doubling bifurcation phenomenon strongly indicates that chaos indeed exist in this system. After finding the chaos phenomenon, this paper continues to investigate the relationship between the decay time of the autocorrelation function and the largest Lyapunov exponent. The Poincaré mapping points are chosen to be the sampled function of the discrete autocorrelation function. It's found that a power model of regression analysis can fit with good accuracy the data points, which are composed of the mapping times for the autocorrelation to decay into the square of the mean of the Poincaré points and the corresponding 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.

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.

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.

A Very Simple Method to Calculate the (Positive) Largest Lyapunov Exponent Using Interval Extensions

2016 ◽  
Vol 26 (13) ◽  
pp. 1650226 ◽  
Author(s):  
Eduardo M. A. M. Mendes ◽  
Erivelton G. Nepomuceno
Keyword(s):  
Lyapunov Exponent ◽  
Lower Bound ◽  
Equations Of Motion ◽  
Original System ◽  
Glass System ◽  
Largest Lyapunov Exponent ◽  
Simple Method ◽  
Interval Extensions

In this letter, a very simple method to calculate the positive Largest Lyapunov Exponent (LLE) based on the concept of interval extensions and using the original equations of motion is presented. The exponent is estimated from the slope of the line derived from the lower bound error when considering two interval extensions of the original system. It is shown that the algorithm is robust, fast and easy to implement and can be considered as alternative to other algorithms available in the literature. The method has been successfully tested in five well-known systems: Logistic, Hénon, Lorenz and Rössler equations and the Mackey–Glass system.

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

scholarly journals Diagnostic of cardio-vascular disease with help of largest Lyapunov exponent of RR-sequences

2000 ◽  
Vol 11 (5) ◽  
pp. 807-814 ◽  
Author(s):  
Alexey N Pavlov ◽  
Natalia B Janson ◽  
Vadim S Anishchenko ◽  
Vladimir I Gridnev ◽  
Pavel Ya Dovgalevsky
Keyword(s):  
Lyapunov Exponent ◽  
Vascular Disease ◽  
Largest Lyapunov Exponent ◽  
Cardio Vascular Disease ◽  
Cardio Vascular
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