STRANGE NONCHAOTIC ATTRACTORS

Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic Attractors (SNAs). Such attractors are generic in quasiperiodically driven nonlinear systems, and like strange attractors, are geometrically fractal. The largest Lyapunov exponent is zero or negative: trajectories do not show exponential sensitivity to initial conditions. In recent years, SNAs have been seen in a number of diverse experimental situations ranging from quasiperiodically driven mechanical or electronic systems to plasma discharges. An important connection is the equivalence between a quasiperiodically driven system and the Schrödinger equation for a particle in a related quasiperiodic potential, showing a correspondence between the localized states of the quantum problem with SNAs in the related dynamical system. In this review we discuss the main conceptual issues in the study of SNAs, including the different bifurcations or routes for the creation of such attractors, the methods of characterization, and the nature of dynamical transitions in quasiperiodically forced systems. The variation of the Lyapunov exponent, and the qualitative and quantitative aspects of its local fluctuation properties, have emerged as an important means of studying fractal attractors, and this analysis finds useful application here. The ubiquity of such attractors, in conjunction with their several unusual properties, suggests novel applications.

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Amplitude Suppression and Chaos Control in Epileptic EEG Signals

Computational and Mathematical Methods in Medicine ◽

10.1080/10273660600890032 ◽

2006 ◽

Vol 7 (1) ◽

pp. 53-66 ◽

Cited By ~ 3

Author(s):

Kaushik Majumdar ◽

Mark H. Myers

Keyword(s):

Lyapunov Exponent ◽

Chaos Control ◽

Initial Conditions ◽

Chaotic Signal ◽

Patient Specific ◽

Largest Lyapunov Exponent ◽

Eeg Signals ◽

Chaotic Dynamical System ◽

Sensitive Dependence ◽

Improve Patient

In this paper we have proposed a novel amplitude suppression algorithm for EEG signals collected during epileptic seizure. Then we have proposed a measure of chaoticity for a chaotic signal, which is somewhat similar to measuring sensitive dependence on initial conditions by measuring Lyapunov exponent in a chaotic dynamical system. We have shown that with respect to this measure the amplitude suppression algorithm reduces chaoticity in a chaotic signal (EEG signal is chaotic). We have compared our measure with the estimated largest Lyapunov exponent measure by the largelyap function, which is similar to Wolf's algorithm. They fit closely for all but one of the cases. How the algorithm can help to improve patient specific dosage titration during vagus nerve stimulation therapy has been outlined.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127494000071 ◽

1994 ◽

Vol 04 (01) ◽

pp. 87-98 ◽

Cited By ~ 14

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.

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LOW-DIMENSIONAL BEHAVIOR OF NONLINEAR TEARING MODE DYNAMICS IN A ROTATING PLASMA

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493000957 ◽

1993 ◽

Vol 03 (05) ◽

pp. 1155-1168 ◽

Cited By ~ 1

Author(s):

M. PERSSON ◽

C. R. LAING

Keyword(s):

Lyapunov Exponent ◽

Initial Conditions ◽

Largest Lyapunov Exponent ◽

Tearing Mode ◽

Oscillatory State ◽

The Largest Lyapunov Exponent ◽

Low Dimensional ◽

Sensitivity To Initial Conditions

Simulations of resistive magnetohydrodynamics in a rotating plasma are analyzed by calculating the correlation dimensions and the local intrinsic dimensions. A rotating plasma with a nonlinear oscillatory state is found to be associated with a low-dimensional attractor. The solutions are also checked for sensitivity to initial conditions, characterized by the sign of the largest Lyapunov exponent.

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Hidden Strange Nonchaotic Attractors

Mathematics ◽

10.3390/math9060652 ◽

2021 ◽

Vol 9 (6) ◽

pp. 652

Author(s):

Marius-F. Danca ◽

Nikolay Kuznetsov

Keyword(s):

Finite Time ◽

Initial Conditions ◽

Power Spectra ◽

Recurrence Plot ◽

Chaotic Attractors ◽

Largest Lyapunov Exponent ◽

Hidden Attractor ◽

Test Power ◽

Finite Time Lyapunov Exponents ◽

Strange Nonchaotic Attractors

In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic attractor of the Rabinovich–Fabrikant system are comparatively analyzed.

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Largest Lyapunov Exponent and Attractors Applied to Stability Study of Nuclear Reactors

10.26678/abcm.cobem2017.cob17-0653 ◽

2017 ◽

Author(s):

WILMER ARUQUIPA COLOMA ◽

Antonella Lombardi Costa ◽

Claubia Pereira

Keyword(s):

Lyapunov Exponent ◽

Nuclear Reactors ◽

Stability Study ◽

Largest Lyapunov Exponent

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EFFECTS OF TREND AND PERIODICITY ON THE CORRELATION DIMENSION AND THE LYAPUNOV EXPONENTS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408022640 ◽

2008 ◽

Vol 18 (12) ◽

pp. 3679-3687 ◽

Cited By ~ 6

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.

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

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.197.271 ◽

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

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.342-343.581 ◽

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.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416502266 ◽

2016 ◽

Vol 26 (13) ◽

pp. 1650226 ◽

Cited By ~ 20

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.

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