• Journal of the Korean Data and Information Science Society
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• 제23권3호
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• pp.595-604
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• 2012

In this article, we consider chaotic behavior happened in nonsmooth dynamical systems. To quantify such a behavior, a computation of Lyapunov exponents for chaotic orbits of a given nonsmooth dynamical system is focused. The Lyapunov exponent is a very important concept in chaotic theory, because this quantity measures the sensitive dependence on initial conditions in dynamical systems. Therefore, Lyapunov exponents can decide whether an orbit is chaos or not. To measure the sensitive dependence on initial conditions for nonsmooth dynamical systems, the calculation of Lyapunov exponent plays a key role, but in a theoretical point of view or based on the definition of Lyapunov exponents, Lyapunov exponents of nonsmooth orbit could not be calculated easily, because the Jacobian derivative at some point in the orbit may not exists. We use an algorithmic calculation method for computing Lyapunov exponents using time series for a two dimensional piecewise smooth dynamic system.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 2001년도 ICCAS
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• pp.43.1-43
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• 2001

We propose two control methods of the Lyapunov exponents for Jordan-type recurrent neural networks. Both the two methods are formulated by a gradient-based learning method. The first method is derived strictly from the definition of the Lyapunov exponents that are represented by the state transition of the recurrent networks. The first method can control the complete set of the exponents, called the Lyapunov spectrum, however, it is computationally expensive because of its inherent recursive way to calculate the changes of the network parameters. Also this recursive calculation causes an unstable control when, at least, one of the exponents is positive, such as the largest Lyapunov exponent in the recurrent networks with chaotic dynamics. To improve stability in the chaotic situation, we propose a non recursive formulation by approximating ...

• Structural Engineering and Mechanics
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• 제52권3호
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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.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 1997년도 한국자동제어학술회의논문집; 한국전력공사 서울연수원; 17-18 Oct. 1997
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• pp.1844-1847
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• 1997

Assuming that EEG(electroencephalogram), which is generated by a nonlinear electrical of billions of neurons in the brain, has chaotic characteristics, it is confirmend by frequency spectrum analysis, log frequency spectrum analysis, correlation dimension analysis and Lyapunov exponents analysis. Some chaotic characteristics are related to the degree of brain activity. The slope of log frequency spectrum increases and the correlation dimension decreasess with respect to the activities, while the largest Lyapunov exponent has only a rough correlation.

• 충청수학회지
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• 제25권3호
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• pp.589-597
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• 2012

In this paper we investigate the stability of solutions of the perturbed differential equations with the positive order of the perturbation by using the notion of the Lyapunov exponent of unperturbed equations and an integral inequality of Bihari type.

• 한국공작기계학회:학술대회논문집
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• 한국공작기계학회 1996년도 추계학술대회 논문
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• pp.96-100
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• 1996

So far the analysis and modeling of cutting process is studied commonly assumed as being linear stochastic or chaotic without experimental verification. So we verified force signals of cutting process(ball end-milling) is low-dimensional chaos by calculating Lyapunov Exponents. reconstructing attractor using time delay coordinates and calcula-ting it's fractal dimension.

• 한국정밀공학회지
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• 제16권1호통권94호
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• pp.150-160
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• 1999

본 논문은 ‘파트 1’에 그 기초를 두었으며, 실제 실험 상황에의 응용예를 들었다. 보편적인 ‘이중-우물 위치 진동기(double-well potential vibrator)'를 외부 공기압 감쇠기를 장치할 수 있도록 수정하였다. 감쇠는 높음 또는 낮음 으로 조정할 수 있도록 하였다. 이 실험계는 주기운동부터 카오스 운동까지 다양한 동적 특성을 보여준다. 힘-위상(Force-Stare Mapping) 방법이 선형상태 및 카오스상태에 응용되었으며, 특히 감쇠의 높고 낮음의 파악에 그 중점을 두었다. 그리고 , 부분발산지수들(Short term averaged Lyapunov exponents)의 합이 또한 감쇠를 파악함과 동시에 높은 감쇠에서 낮은 감쇠로의 변화를 감시할 수 있음을 보였다. 이 두가지 방법들을 비교하였으며 논하였다.

• Nuclear Engineering and Technology
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• 제48권2호
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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.

• 한국정밀공학회지
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• 제16권1호통권94호
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• pp.141-149
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• 1999

어느 한 계가 양수의 발산지수(Lyapunov exponent)를 가질 때 이 계는 카오스계로 분류되며 그 동특성은 예측이 불가능해 진다. 감쇠 기계계(소산계)에서는 위상공간(phase space)의 초기 부피가 시간에 따라 수축한다. 발산 지수들의 합은 음수이며 그 기계계의 감쇠와 관련되며, 따라서 발산지수들의 합은 감쇠의 변화를 감시하는데 사용되어질 수 있다. 그러나 그 감쇠변화를 감시하기 위해서는 발산지수를 계산하는데 사용하는 신호(data) 부분(segment)이 짧아야 한다. 이는 문제점을 야기시키는데 그 이유는 발산지수가 아주 많은 양의 발산률(divergence rate)의 평균으로서 구해지기 때문이다. 이 문제를 극복하기 위해서, 본 저자는 '순간발산지수(Instantaneous Lyapunov Exponent)'를 도입하였으며, 이 순간발산지수들의 합이 어떻게 기계계의 감쇠와 관련되어지는 가에 대하여 기술하였다. 미분방적식과 시계열(time series)을 이용한 컴퓨터 시뮬레이션은 '순간발산지수들의 합'의 중요성을 입증하였다. 그러나 시계열(또는 실험신호)로 부터의 정확한 순간발산지수를 측정하기는 매우 힘들기 때문에 '부분발산지수(Short term averaged Lyapunov Exponent)'를 또한 도입하였다.

• 대한기계학회논문집B
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• 제21권12호
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• pp.1615-1623
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• 1997

Numerical study on the chaotic stirring of the screw extruder model proposed has been performed. The velocity field was used in obtaining the trajectories of passive particles for studying the stirring effect of the screw extruder. Two nonlinear dynamical tools, that are Poincare sections and Lyapunov exponents, were used in analysing the stirring effect. The Poincare sections and the Lyapunov exponents show that the stirring effect is most satisfactory, when n(the number of flights in a section) is 1, for the case a (aspect ratio ; flight height divided by the spacing between flights) being O.1. It is also required to set n=3, or 5 at a= 0.2, 0.3 for a uniform stirring.