The real Schur decomposition estimates Lyapunov characteristic exponents with multiplicity greater than one

Masarati Pierangelo; Tamer Aykut

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The real Schur decomposition estimates Lyapunov characteristic exponents with multiplicity greater than one

机译：实际Schur分解估计Lyapunov特征指数的多重性大于1

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Lyapunov characteristic exponents are indicators of the nature and of the stability properties of solutions of differential equations. The estimation of Lyapunov exponents of algebraic multiplicity greater than 1 is troublesome. In this work, we intuitively derive an interpretation of higher multiplicity Lyapunov exponents in forms that occur in simple linear time invariant problems of engineering relevance. We propose a method to determine them from the real Schur decomposition of the state transition matrix of the linear, nonautonomous problem associated with the fiducial trajectory. So far, no practical way has been found to formulate the method as an algorithm capable of mitigating over- or underflow in the numerical computation of the state transition matrix. However, this interesting approach in some practical cases is shown to provide quicker convergence than usual methods like the discrete QR and the continuous QR and Singular Value Decomposition (SVD)methods when Lyapunov exponents with multiplicity greater than one are present.

机译：李雅普诺夫特征指数是微分方程解的性质和稳定性的指标。估计大于1的代数多重性的Lyapunov指数很麻烦。在这项工作中，我们直观地得出了以工程相关性的简单线性时不变问题中出现的形式对更高重数Lyapunov指数的解释。我们提出了一种从与基准轨迹相关的线性，非自治问题的状态转移矩阵的实际Schur分解中确定它们的方法。迄今为止，还没有找到将这种方法表示为能够减轻状态转移矩阵的数值计算中的上溢或下溢的算法的实用方法。但是，在存在多个Lyapunov指数的情况下，这种有趣的方法在某些实际情况下显示出比常规方法（例如离散QR和连续QR和奇异值分解（SVD）方法）更快的收敛性。

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来源
《Proceedings of the institution of mechanical engineers》 |2016年第k4期|568-578|共11页
作者
Masarati Pierangelo; Tamer Aykut;
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作者单位

Politecn Milan, Milan, Italy;

Politecn Milan, Milan, Italy;

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收录信息美国《科学引文索引》(SCI);美国《工程索引》(EI);
原文格式 PDF
正文语种 eng
中图分类
关键词
Lyapunov characteristic exponents; multiplicity; real Schur decomposition; stability; dynamics;

机译：李雅普诺夫特征指数多重实Schur分解稳定性动力学;

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机译：利用李雅普诺夫特征指数估计的轨迹稳定性敏感性
2. Lyapunov exponents with multiplicity 1 for deterministic products of matrices [J] . Bonatti C, Viana M Ergodic Theory and Dynamical Systems . 2004,第6期

机译：矩阵确定性乘数为1的Lyapunov指数
3. On the Lyapunov Characteristic Exponents and Lower Perron Exponents of Solutions of Coppel-Conti Linear Differential Systems [J] . N. A. Izobov, R. A. Prokhorova Differential equations: A translation of differensial'nye uraveniya . 2006,第12期

机译：Coppel-Conti线性微分系统解的Lyapunov特征指数和下Perron指数
4. Characterizing the gait dynamic by estimating Lyapunov exponents on gait kinematic trajectories in Parkinson's Disease [C] . David Ricaurte, Gustavo Pineda, Eduardo Romero Conference on International Symposium on Medical Information Processing and Analysis . 2020

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5. On the computation of Lyapunov characteristic exponents. [D] . von Bremen, Hubertus Friedrich Georg. 1999

机译：关于李雅普诺夫特征指数的计算。
6. Characteristic distribution of finite-time Lyapunov exponents for chimera states [O] . André E. Botha -1

机译：嵌合状态的有限时间Lyapunov指数的特征分布
7. EEG Signal Discrimination using Non-linear Dynamics in the EMD Domain S. M. Shafiul Alam,S. M. Shafiul Alam,Aurangozeb, and Syed TarekShahriar Abstract—An EMD-chaos based approach is proposed todiscriminate EEG signals corresponding to healthy persons,and epileptic patients during seizure-free intervals and seizureattacks. An electroencephalogram (EEG) is first empiricallydecomposed to intrinsic mode functions (IMFs). The nonlineardynamics of these IMFs are quantified in terms of the largestLyapunov exponent (LLE) and correlation dimension (CD).This chaotic analysis in EMD domain is applied to a large groupof EEG signals corresponding to healthy persons as well asepileptic patients (both with and without seizure attacks). It isshown that the values of the obtained LLE and CD exhibitfeatures by which EEG for seizure attacks can be clearlydistinguished from other EEG signals in the EMD domain.Thus, the proposed approach may aid researchers in developingeffective techniques to predict seizure activities. Index Terms—Electroencephalogram (EEG), empiricalmode decomposition (EMD), largest Lyapunov exponent (LLE),correlation dimension (CD), epileptic seizures. The Authors are with the Electrical and Electronic EngineeringDepartment, Bangladesh University of Engineering and Technology,Dhaka-1000, Bangladesh (e-mail: imamul@eee.buet.ac.bd) PDF Cite: S. M. Shafiul Alam,S. M. Shafiul Alam,Aurangozeb, and Syed Tarek Shahriar, "EEG Signal Discrimination using Non-linear Dynamics in the EMD Domain," International Journal of Computer and Electrical Engineering vol. 4, no. 3, pp. 326-330, 2012. PREVIOUS PAPER Perception of Emotions Using Constructive Learningthrough Speech NEXT PAPER Physical Layer Impairments Aware OVPN Connection Selection Mechanisms Copyright © 2008-2013. International Association of Computer Science and Information Technology Press (IACSIT Press) [O] . S. M. Shafiul Alam, Syed TarekShahriar 2012

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8. Lyapunov Exponents and Rotation Numbers of Linear Systems with Real Noise [R] . Pinsky, M., Wihstutz, V. 1990

机译：具有实噪声的线性系统的Lyapunov指数和旋转数

The real Schur decomposition estimates Lyapunov characteristic exponents with multiplicity greater than one

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