EXPERIMENTAL EVIDENCE OF DIFFUSIVE DYNAMICS AND “RANDOM WALKING” IN A SIMPLE DETERMINISTIC MECHANICAL SYSTEM: THE SHAKEN PENDULUM

1992 ◽  
Vol 02 (04) ◽  
pp. 983-988 ◽  
Author(s):  
PHILIP V. BAYLY ◽  
LAWRANCE N. VIRGIN
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Power Spectrum ◽  
Chaotic Dynamics ◽  
Angular Displacement ◽  
Translational Symmetry ◽  
Random Walking ◽  
Diffusive Dynamics ◽  
Nonlinear Maps ◽  
Deterministic Diffusion

An experimental model of a simple pendulum, harmonically shaken, displays chaotic dynamics. Moreover, in strongly excited chaotic regimes the time series of total angular displacement, which is rarely examined, wanders unboundedly, displaying a power spectrum which falls off as 1/fα over several decades. This behavior corresponds to deterministic diffusion, which has been found in simulations of nonlinear maps with periodic translational symmetry. The displacement time series obtained by sampling the pendulum displacement once per cycle is self-affine and quantitatively similar to Brownian motion.

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.

Online Monitoring Recognition Theory Based on the Time Series of Chatter

2013 ◽  
Vol 819 ◽  
pp. 160-164
Author(s):  
Yong Xiang Jiang ◽  
Bing Du ◽  
Pan Zhang ◽  
San Peng Deng ◽  
Yu Ming Qi
Keyword(s):  
Time Series ◽  
Chaotic Dynamics ◽  
Online Monitoring ◽  
Vibration Signal ◽  
Threshold Value ◽  
Coarse Grained ◽  
Attractor Dimension ◽  
Machining Chatter ◽  
Recognition Theory ◽  
On Line

On-line monitoring recognition for machining chatter is one of the key technologies in manufacturing. Based on the nonlinear chaotic control theory, the vibration signal discrete time series for on-line monitoring indicator is studed. As in chatter the chaotic dynamics process attractor dimension is reduced, the KolmogorovSinai entropy (K-S) index is extracted to reflected the regularity of workpiece chatter, then the k-S entropy is simplified by coarse - grained entropy rate (CER), which can easily evaluated as chatter online monitoring threshold value. The milling test shows that the CER have a sharp decline when chatter occurre, and can quickly and accurately forecast chatter.

Gear Damage Detection Using Chaotic Dynamics Techniques: A Preliminary Study

1995 ◽  
Author(s):  
M. Farid Golnaraghi ◽  
DerChyan Lin ◽  
Paul Fromme
Keyword(s):  
Time Series ◽  
Time Series Analysis ◽  
Damage Detection ◽  
Correlation Dimension ◽  
Chaotic Dynamics ◽  
Crack Detection ◽  
Gear Tooth ◽  
Vibration Signal ◽  
Nonlinear Time Series Analysis ◽  
Preliminary Study

Abstract This paper is a preliminary study applying nonlinear time series analysis to crack detection in gearboxes. Our investigations show that the vibration signal emerging from a gearbox is chaotic. Appearance of a crack in a gear tooth alters this response and hence the chaotic signature. We used correlation dimension and Lyapunov exponents to quantify this change. The main goal of this study is to point out the great potential of these methods in detection of cracks and faults in machinery.

scholarly journals Some Applications of Lévy Processes to Stochastic Investment Models for Actuarial Use

1998 ◽  
Vol 28 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Continuous Time ◽  
Investment Model ◽  
Stable Processes ◽  
Gamma Processes ◽  
Investment Models

AbstractThis paper presents a continuous time version of a stochastic investment model originally due to Wilkie. The model is constructed via stochastic differential equations. Explicit distributions are obtained in the case where the SDEs are driven by Brownian motion, which is the continuous time analogue of the time series with white noise residuals considered by Wilkie. In addition, the cases where the driving “noise” are stable processes and Gamma processes are considered.

Experimental Techniques for Controlling Chaos in Lasers

1998 ◽  
Vol 08 (09) ◽  
pp. 1759-1768 ◽  
Author(s):  
R. Meucci ◽  
A. Labate ◽  
M. Ciofini
Keyword(s):  
Power Spectrum ◽  
Chaotic Dynamics ◽  
Feedback Loop ◽  
Control Parameter ◽  
Experimental Techniques ◽  
Controlling Chaos ◽  
Sinusoidal Modulation ◽  
Control Schemes ◽  
Frequency Components

This paper presents two control schemes for the chaotic dynamics of CO 2 laser with feedback which can be applied after the recognition of a leading frequency of the motion in the power spectrum. The first one is realized by means of a selective feedback loop which rejects all the frequency components except that of the leading cycle to be stabilized. The second one consists in a resonant sinusoidal modulation of the control parameter.

scholarly journals Analysis of Dynamical Behavior for Epidemic Disease COVID-19 with Application

2021 ◽  
Vol 12 (4) ◽  
pp. 568-577
Author(s):  
Dr. Maysoon M. Aziz, Et. al.
Keyword(s):  
Time Series ◽  
Linear System ◽  
Power Spectrum ◽  
Dynamical Behavior ◽  
Simulated Data ◽  
Real Data ◽  
Sir Model ◽  
Epidemic Disease ◽  
The World ◽  
Non Linear System

In this paper, we will use the differential equations of the SIR model as a non-linear system, by using the Runge-Kutta numerical method to calculate simulated values for known epidemiological diseases related to the time series including the epidemic disease COVID-19, to obtain hypothetical results and compare them with the dailyreal statisticals of the disease for counties of the world and to know the behavior of this disease through mathematical applications, in terms of stability as well as chaos in many applied methods. The simulated data was obtained by using Matlab programms, and compared between real data and simulated datd were well compatible and with a degree of closeness. we took the data for Italy as an application.  The results shows that this disease is unstable, dissipative and chaotic, and the Kcorr of it equal (0.9621), ,also the power spectrum system was used as an indicator to clarify the chaos of the disease, these proves that it is a spread,outbreaks,chaotic and epidemic disease .

scholarly journals Time series analysis of trial-to-trial variability of MEG power spectrum during rest state, unattented listening and frequency-modulated tones classification

2021 ◽  
Author(s):  
Lech Kipiński ◽  
Wojciech Kordecki
Keyword(s):  
Stochastic Process ◽  
Time Series ◽  
Power Spectrum ◽  
Spectral Density ◽  
Cognitive Task ◽  
Clinical Neuroscience ◽  
Rest State ◽  
Short Time ◽  
Low Complex ◽  
Dominant Frequencies

AbstractThe nonstationarity of EEG/MEG signals is important for understanding the functioning of human brain. From the previous research we know that even very short, i.e. 250—500ms MEG signals are variance-nonstationary. The covariance of stochastic process is mathematically associated with its spectral density, therefore we investigate how the spectrum of such nonstationary signals varies in time.We analyze the data from 148-channel MEG, that represent rest state, unattented listening and frequency-modulated tones classification. We transform short-time MEG signals to the frequency domain using the FFT algorithm and for the dominant frequencies 8—12 Hz we prepare the time series representing their trial-to-trial variability. Then, we test them for level- and trend-stationarity, unit root, heteroscedasticity and gaussianity and based on their properties we propose the ARMA-modelling for their description.The analyzed time series have the weakly stationary properties independently of the functional state of brain and localization. Only their small percentage, mostly related to the cognitive task, still presents nonstationarity. The obtained mathematical models show that the spectral density of analyzed signals depends on only 2—3 previous trials.The presented method has limitations related to FFT resolution and univariate models, but it is not computationally complicated and allows to obtain a low-complex stochastic models of the EEG/MEG spectrum variability.Although the physiological short-time MEG signals are in principle nonstationary in time domain, its power spectrum at the dominant frequencies varies as weakly stationary stochastic process. Described technique has the possible applications in prediction of the EEG/MEG spectral properties in theoretical and clinical neuroscience.

scholarly journals Wavelet analysis applied on temporal data sets in order to reveal possible pre-seismic radio anomalies and comparison with the trend of the raw data 

2021 ◽  
Author(s):  
Giovanni Nico ◽  
Pier Francesco Biagi ◽  
Anita Ermini ◽  
Mohammed Yahia Boudjada ◽  
Hans Ulrich Eichelberger ◽  
...  
Keyword(s):  
Time Series ◽  
Wavelet Analysis ◽  
Power Spectrum ◽  
Power Spectra ◽  
Wavelet Spectrum ◽  
Time Signal ◽  
Time Behavior ◽  
Sudden Increase ◽  
Night Time ◽  
Raw Data

<p>Since 2009, several radio receivers have been installed throughout Europe in order to realize the INFREP European radio network for studying the VLF (10-50 kHz) and LF (150-300 kHz) radio precursors of earthquakes. Precursors can be related to “anomalies” in the night-time behavior of  VLF signals. A suitable method of analysis is the use of the Wavelet spectra.  Using the “Morlet function”, the Wavelet transform of a time signal is a complex series that can be usefully represented by its square amplitude, i.e. considering the so-called Wavelet power spectrum.</p><p>The power spectrum is a 2D diagram that, once properly normalized with respect to the power of the white noise, gives information on the strength and precise time of occurrence of the various Fourier components, which are present in the original time series. The main difference between the Wavelet power spectra and the Fourier power spectra for the time series is that the former identifies the frequency content along the operational time, which cannot be done with the latter. Anomalies are identified as regions of the Wavelet spectrogram characterized by a sudden increase in the power strength.</p><p>On January 30, 2020 an earthquake with Mw= 6.0 occurred in Dodecanese Islands. The results of the Wavelet analysis carried out on data collected some INFREP receivers is compared with the trends of the raw data. The time series from January 24, 2020 till January 31, 2000 was analyzed. The Wavelet spectrogram shows a peak corresponding to a period of 1 day on the days before January 30. This anomaly was found for signals transmitted at the frequencies 19,58 kHz, 20, 27 kHz, 23,40 kHz with an energy in the peak increasing from 19,58 kHz to 23,40 kHz. In particular, the signal at the frequency 19,58 kHz, shows a peak on January 29, while the frequencies 20,27 kHz and 23,40 kHz are characterized by a peak starting on January 28 and continuing to January 29. The results presented in this work shows the perspective use of the Wavelet spectrum analysis as an operational tool for the detection of anomalies in VLF and LF signal potentially related to EQ precursors.</p>

LONG RANGE CORRELATION PROPERTIES OF AFTERSHOCK RELAXATION SIGNALS

Fractals ◽  
1995 ◽  
Vol 03 (04) ◽  
pp. 839-847 ◽  
Author(s):  
A. VESPIGNANI ◽  
A. PETRI ◽  
A. ALIPPI ◽  
G. PAPARO ◽  
M. COSTANTINI
Keyword(s):  
Time Series ◽  
Acoustic Emission ◽  
Statistical Analysis ◽  
Power Spectrum ◽  
Power Law ◽  
Amplitude Distribution ◽  
Relaxation Processes ◽  
Critical Dynamics ◽  
Range Correlation ◽  
Correlation Properties

Relaxation processes taking place after microfracturing of laboratory samples give rise to ultrasonic acoustic emission signals. Statistical analysis of the resulting time series has revealed many features which are characteristic of critical phenomena. In particular, the autocorrelation functions obey a power-law behavior, implying a power spectrum of the kind 1/f. Also the amplitude distribution N(V) of such signals follows a power law, and the obtained exponents are consistent with those found in other experiments: N(V) dV≃V–γ dV, with γ=1.7±0.2. We also analyzed the distribution N(τ) of the delay time τ between two consecutive acoustic emission events. We found that a N(τ) distribution rather close to a power law constitutes a common feature of all the recorded signals. These experimental results can be considered as a striking evidence for a critical dynamics underlying the microfracturing processes.

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