• Transactions on Control, Automation and Systems Engineering
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• 제4권2호
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• pp.124-129
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• 2002

This paper demonstrates that the largest Lyapunov exponent λ of recurrent neural networks can be controlled efficiently by a stochastic gradient method. An essential core of the proposed method is a novel stochastic approximate formulation of the Lyapunov exponent λ as a function of the network parameters such as connection weights and thresholds of neural activation functions. By a gradient method, a direct calculation to minimize a square error (λ - λ$\^$obj/)$^2$, where λ$\^$obj/ is a desired exponent value, needs gradients collection through time which are given by a recursive calculation from past to present values. The collection is computationally expensive and causes unstable control of the exponent for networks with chaotic dynamics because of chaotic instability. The stochastic formulation derived in this paper gives us an approximation of the gradients collection in a fashion without the recursive calculation. This approximation can realize not only a faster calculation of the gradient, but also stable control for chaotic dynamics. Due to the non-recursive calculation. without respect to the time evolutions, the running times of this approximation grow only about as N$^2$ compared to as N$\^$5/T that is of the direct calculation method. It is also shown by simulation studies that the approximation is a robust formulation for the network size and that proposed method can control the chaos dynamics in recurrent neural networks efficiently.

• 대한기계학회논문집
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• 제18권2호
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• pp.380-388
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• 1994

Study on the chaotic stirring has been performed numerically and experimentally for a shallow rectangular tank accompanying a vortex shedding. The model is composed of a rectangular tank with a vertical plate with a length half the width of the tank. The tank is subject to a horizontal sinusoidal oscillation. The chaotic stirring was analysed by Poincare sections, unstable manifolds and Lyapunov exponents. As Reynolds number is increased the stirring effect is decreased due to the growth of a regular regions near the lower surface of the tank. In the other hand decrease of Reynolds number gives a weaker vortex shedding resulting in the poorer stirring effect. It was also found that the Lyapunov exponent is the highest at the dimensionless period of 1.3-1.5, which seems to be the best condition for the efficient stirring. The experimental visualization for the deformation of materials exhibits the striation pattern similar to the unstable manifold obtained numerically.

• 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.

• 대한의용생체공학회:의공학회지
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• 제15권3호
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• pp.237-244
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• 1994

1Hz에서 20Hz까지의 단속 주파수를 지닌 청각자극을 가해 얻은 EEG 신호에서 자극에 따른 신호의 정성적이고 정량적인 특성을 카오스 분석방법을 통해 밝혔다. 먼저, 뇌전위 신호에 전반적으로 나타나는 일반적인 카오스 특징(fractal mechanism, I/f frequency spectrum, positive Lyapunov exponent 등등)을 확인하였다. 유발전위에 대해서는 자극의 주파수에 따른 주기배증을 경유한 카오스로 가는 길(route to chaos)과 2차원 pseudo-Phase portrait의 뿌앙까레 단면에서의 기하학적 모양(topological property)의 변화를 관찰하였고, 자발전위가 포함된 유발전위에 대해서는 적절한 bases를 지닌 3차원 phase space에서 기이한 끌개(chaotic attractor)가, 유발전위의 정보를 지닌채 보여졌다. 끝으로 자극 주파수(단속 주파수)변화와 측정이 이루어진 머리표면에서의 공간적 위치에 따른 Lyapunov exponent값 변화를 의미있게 해석하였다. 이 결과는 무질서하게 보이는 뇌전위신호에서 주어진 청각자극에 대한 정보를 얻는 새로운 방법을 제시하게 된다.

• 한국지능시스템학회:학술대회논문집
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• 한국퍼지및지능시스템학회 2003년도 추계 학술대회 학술발표 논문집
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• pp.110-113
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• 2003

In this paper, we propose that the chaotic behavior analysis in the mobile robot of embedding Arnold 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.

• 한국지능시스템학회:학술대회논문집
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• 한국퍼지및지능시스템학회 2003년도 추계 학술대회 학술발표 논문집
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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

• International Journal of Automotive Technology
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• 제8권4호
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• pp.467-476
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• 2007

This work verifies the chaotic motion of a steer-by-wire vehicle dynamic system, and then elucidates an application of dither smoothing to control the chaos of a vehicle model. The largest Lyapunov exponent is estimated from the synchronization to identify periodic and chaotic motions. Then, a bifurcation diagram reveals complex nonlinear behaviors over a range of parameter values. Finally, a method for controlling a chaotic vehicle dynamic system is proposed. This method involves applying another external input, called a dither signal, to the system. The designed controller is demonstrated to work quite well for nonlinear systems in achieving robust stability and protecting the vehicle from slip or spin. Some simulation results are presented to establish the feasibility of the proposed method.

• 대한기계학회:학술대회논문집
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• 대한기계학회 2007년도 춘계학술대회A
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• pp.792-795
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• 2007

To quantify irregular body motions the time series analysis was applied to the gait study. The motions obtained from gait experiment are complex to exhibit nonlinear behaviors. The purpose of this study is to measure quantitatively the characteristics of the major six joints of the body during walking. The gait experiments were carried out for eighteen young males walking on a motor driven treadmill. Joint motions were captured using eight video cameras, and then three dimensional kinematics of the neck and the upper and lower extremities were computed by KWON 3D motion analysis software. The largest Lyapunov exponent was calculated from the time series to quantify stabilities of each joint. The results provides a data set of nonlinear dynamic characteristics for six joints engaged in normal walking.

• 대한수학회보
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• 제51권2호
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• pp.457-478
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• 2014

In this paper, a 3D autonomous system, which has only stable or non-hyperbolic equilibria but still generates chaos, is presented. This system is topologically non-equivalent to the original Lorenz system and all Lorenz-type systems. This motivates us to further study some of its dynamical behaviors, such as the local stability of equilibrium points, the Lyapunov exponent, the dissipativity, the chaotic waveform in time domain, the continuous frequency spectrum, the Poincar$\acute{e}$ map and the forming mechanism for compound structure of its special cases. Especially, with the help of the Project Method, its Hopf bifurcation of codimension one is in detailed formulated. Numerical simulation results not only examine the corresponding theoretical analytical results, but also show that this system possesses abundant and complex dynamical properties not solved theoretically, which need further attention.