• Annals of Clinical Neurophysiology
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• v.8 no.2
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• pp.135-145
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• 2006

Background: Parkinson's disease is movement disorder due to dopaminergic deficiency. It has been noted that cognitive dysfunction also presented on Parkinson's disease patients. But, it is not clear whether such a cognitive dysfunction was a dopaminergic dysfunction or cholinergic dysfunction. Using linear and non-linear analyses, we analysed the effect of cognitive and motor symptom on EEG change. Methods: EEGs were recorded from patients with Parkinson's disease and essential tremor, and normal controls during rest. We calculated the power spectrum, correlation dimension and Lyapunov exponent by using 'Complexity'program. The power spectrum, correlation dimension, and Lyapunov exponent were compared between Parkinson's disease patients and essential tremor patients. Results: Theta power was increased in Parkinson's disease patient group. Correlation dimension was increased in Parkinson's disease patients. Positive correlation was noted between MMSE and correlation dimension, and negative correlation was noted between MMSE and Lyapunov exponent. Lyapunov exponent was decreased in Parkinson's disease patient. Conclusions: We conclude that the state of Parkinson's disease patient is characterized by increased correlation dimension and decreased Lyapunov exponent.

• The Pure and Applied Mathematics
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• v.7 no.2
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• pp.87-100
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• 2000

In this paper, we try to approach chasos with numerical method. After investigating nonlinear dynamcis (chaos) theory, we introduce Lyapunov exponent as chaos\`s index. To look into the existence of chaos in 2-dimensional difference equation we computes Lypunov exponent and examine the various behaviors of solutions by difurcation map.

• Annals of Clinical Neurophysiology
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• v.3 no.1
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• pp.31-36
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• 2001

Background & Objectives : Fractal Dimension(FD) could be an index of correlation between variable parameters in non-linear chaotic signals. We tried to demonstrate that EEG wave is compatible with chaotic waves by measuring the Lyapunov exponent index and compared the difference of FD between variable age groups(teens, 30's, 50's) Methods : We estimated the Lyapunov exponent index and the FD from digital EEG data among five persons in each normal age groups by using the software which is programmed in our laboratory. Statistical analysis was done with SPSS win 8.0. The statistical differences of Lyapunov exponent index and FD between each electrodes and each age groups were done with ANOVA and paired sample t-test. Result : The Lyapunov exponent indexes were larger than 1 in each electrode and age group. There is no statistical difference in FD between each electrodes and each age groups. Except in 30th age group. In this group the FD of right hemisphere is larger than that of left hemisphere. Conclusion : The result of Lyapunov exponent index means EEG wave is a non-linear chaotic signal. And the results of FD suggest that chaotic parameters of right hemisphere is larger than those of left hemisphere at rest at least in younger people. We think that chaotic parameters can be a useful tool in investigating the variable diseases or brain states.

• Structural Engineering and Mechanics
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• v.52 no.3
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• pp.525-540
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• 2014

The Lyapunov exponent and moment Lyapunov exponents of Hill's equation with frequency and damping coefficient fluctuated by correlated wideband random processes are studied in this paper. The method of stochastic averaging, both the first-order and the second-order, is applied. The averaged $It\hat{o}$ differential equation governing the pth norm is established and the pth moment Lyapunov exponents and Lyapunov exponent are then obtained. This method is applied to the study of the almost-sure and the moment stability of the stationary solution of the thin simply supported beam subjected to time-varying axial compressions and damping which are small intensity correlated stochastic excitations. The validity of the approximate results is checked by the numerical Monte Carlo simulation method for this stochastic system.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.445-454
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• 2006

We describe a solution to the Ornstein-Uhlenbeck equation $\frac{dI}{dt}-\frac{1}{\tau}$I(t)=cV(t) where V(t) is a constant multiple of a Gaussian white noise. Our solution is based on a discrete set of Gaussian white noise obtained by taking sample points from a sum of single frequency harmonics that have random amplitudes, random frequencies, and random phases. Hence, it is different from the solution by the standard random walk using random numbers generated by the Box-Mueller algorithm. We prove that the power of the signal has the additive property, from which we derive that the Lyapunov characteristic exponent for our solution is positive. This compares with the solution by other methods where the noise is kept to be in an error range so that its Lyapunov exponent is negative.

• Nuclear Engineering and Technology
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• v.48 no.2
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• pp.434-447
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• 2016

The aim of this paper is the study of instability state of boiling water reactors with a method based in largest Lyapunov exponents (LLEs). Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the LLE. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. This method was applied to a set of signals from several nuclear power plant (NPP) reactors under commercial operating conditions that experienced instabilities events, apparently each of a different nature. Laguna Verde and Forsmark NPPs with in-phase instabilities, and Cofrentes NPP with out-of-phases instability. This study presents the results of intrinsic instability in the boiling water reactors of three NPPs. In the analyzed cases the limit cycle was not reached, which implies that the point of equilibrium exerts influence and attraction on system evolution.

• The Journal of the Korea institute of electronic communication sciences
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• v.1 no.1
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• pp.49-55
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• 2006

Applied by periodic Stimulating Currents in Bonhoeffer -Van der Pol(BVP) model, chaotic and periodic phenomena occured at specific conditions. The conditions of the chaotic motion in BVP comprised 0.7182< $A_1$ <0.792 and 1.09< $A_1$ <1.302 proved by the analysis of phase plane, bifurcation diagram, and lyapunov exponent. To control the chaotic motion, two methods were suggested by the first used the amplitude parameter A1, $A1={\varepsilon}((x-x_s)-(y-y_s))$ and the second used the temperature parameterc, $c=c(1+{\eta}cos{\Omega}t)$ which the values of ${\eta},{\Omega}$ varied respectlvly, and $x_s$, $y_s$ are the periodic signal. As a result of simulating these methods, the chaotic phenomena was controlled with the periodic motion of periodisity. The feasibilities of the chaotic and the periodic phenomena were analysed by phase plane Poincare map and lyapunov exponent.

• Journal of Multimedia Information System
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• v.8 no.2
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• pp.131-142
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• 2021

This paper traces the process of constructing a new one-dimensional chaotic map, and will provide a simple application in color image encryption. The use of Sarkovskii's theorem will make it possible to determine the existence of chaos and restrict all conditions to ensure the existence of this new sequence. In addition, the sensitivity to initial conditions will be proved by Lyapunov's index value. Similarly, the performance of this new chaotic map will be illustrated graphically and compared with other chaotic maps most commonly used in cryptography. Finally, a humble color image encryption application will show the power of this new chaotic map.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2003.09b
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• pp.5-8
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• 2003

In this paper, we propose that the chaotic behavior analysis in the mobile robot of embedding Chua's equation with obstacle. In order to analysis of chaotic behavior in the mobile robot, we apply not only qualitative analysis such as time-series, embedding phase plane, but also quantitative analysis such as Lyapunov exponent in the mobile robot with obstacle. In the obstacle, we only assume that all obstacles in the chaos trajectory surface in which robot workspace has an unstable limit cycle with Van der Pol equation

• Journal of computational fluids engineering
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• v.11 no.4 s.35
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• pp.101-106
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• 2006

In the microfluidic devices the most important thing is mixing efficiency ol various fluids. In this study a newly designed miler is proposed to enhance the mixing effect with the purpose to apply it to microchannel mixing in a short future. This design is composed of a channel with cross baffles periodically arranged on the both bottom and top surfaces ol the channel. To obtain the yow patterns, the numerical computation was performed by using a commercial code, ANSYS CFX 10.0. To evaluate the mixing performance, we computed Lyapunov exponent and obtained Poincare sections. it was shown that our design provides the excellent mixing effect.