Lyapunov and marginal instability in Hamiltonian systems

1999 ◽  
Vol 77 (8) ◽  
pp. 603-633 ◽  
Author(s):  
J Grindlay
Keyword(s):  
Lyapunov Exponents ◽  
Nonlinear Oscillator ◽  
Linear Chain ◽  
Mathieu Equation ◽  
Singular Values ◽  
Space Velocity ◽  
Singular Value ◽  
The Stability ◽  
Value Decomposition ◽  
Simple Systems

The variational equations and the evolution matrix are introduced and used to discuss the stability of a bound Hamiltonian trajectory. Singular-value decomposition is applied to the evolution matrix. Singular values and Lyapunov exponents are defined and their properties described. The singular-value expansion of the phase-space velocity is derived. Singular values and Lyapunov exponents are used to characterize the stability behaviour of five simple systems, namely, the nonlinear oscillator with cubic anharmonicity, the quasi-periodic Mathieu equation, the Hénon-Heilesmodel, the 4+2 linear chain with cubic anharmonicity, and an integrable system of arbitrary order.PACS Nos.: 03.20, 05.20

scholarly journals Through-Wall Image Enhancement Based on Singular Value Decomposition

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Muhammad Mohsin Riaz ◽  
Abdul Ghafoor
Keyword(s):  
Singular Value Decomposition ◽  
Image Enhancement ◽  
Visual Inspection ◽  
Signal To Noise Ratio ◽  
Singular Values ◽  
Singular Value ◽  
Information Theoretic ◽  
Value Decomposition ◽  
Suppress Noise ◽  
Wall Imaging

Singular value decomposition and information theoretic criterion-based image enhancement is proposed for through-wall imaging. The scheme is capable of discriminating target, clutter, and noise subspaces. Information theoretic criterion is used with conventional singular value decomposition to find number of target singular values. Furthermore, wavelet transform-based denoising is performed (to further suppress noise signals) by estimating noise variance. Proposed scheme works also for extracting multiple targets in heavy cluttered through-wall images. Simulation results are compared on the basis of mean square error, peak signal to noise ratio, and visual inspection.

scholarly journals Toward efficacy of piecewise polynomial truncated singular value decomposition algorithm in moving force identification

2019 ◽  
Vol 22 (12) ◽  
pp. 2687-2698 ◽  
Author(s):  
Zhen Chen ◽  
Lifeng Qin ◽  
Shunbo Zhao ◽  
Tommy HT Chan ◽  
Andy Nguyen
Keyword(s):  
Singular Value Decomposition ◽  
Singular Values ◽  
Singular Value ◽  
Decomposition Algorithm ◽  
Identification Accuracy ◽  
Piecewise Polynomial ◽  
Truncated Singular Value Decomposition ◽  
Force Identification ◽  
Moving Force ◽  
Value Decomposition

This article introduces and evaluates the piecewise polynomial truncated singular value decomposition algorithm toward an effective use for moving force identification. Suffering from numerical non-uniqueness and noise disturbance, the moving force identification is known to be associated with ill-posedness. An important method for solving this problem is the truncated singular value decomposition algorithm, but the truncated small singular values removed by truncated singular value decomposition may contain some useful information. The piecewise polynomial truncated singular value decomposition algorithm extracts the useful responses from truncated small singular values and superposes it into the solution of truncated singular value decomposition, which can be useful in moving force identification. In this article, a comprehensive numerical simulation is set up to evaluate piecewise polynomial truncated singular value decomposition, and compare this technique against truncated singular value decomposition and singular value decomposition. Numerically simulated data are processed to validate the novel method, which show that regularization matrix [Formula: see text] and truncating point [Formula: see text] are the two most important governing factors affecting identification accuracy and ill-posedness immunity of piecewise polynomial truncated singular value decomposition.

Singular value decomposition for cross‐well tomography

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1655-1661 ◽  
Author(s):  
Reinaldo J. Michelena
Keyword(s):  
Singular Value Decomposition ◽  
Null Space ◽  
Vertical Component ◽  
Singular Values ◽  
Singular Value ◽  
Velocity Estimation ◽  
One Dimensional ◽  
Anisotropic Models ◽  
Value Decomposition ◽  
Frequency Variations

I perform singular value decomposition (SVD) on the matrices that result in tomographic velocity estimation from cross‐well traveltimes in isotropic and anisotropic media. The slowness model is parameterized in four ways: One‐dimensional (1-D) isotropic, 1-D anisotropic, two‐dimensional (2-D) isotropic, and 2-D anisotropic. The singular value distribution is different for the different parameterizations. One‐dimensional isotropic models can be resolved well but the resolution of the data is poor. One‐dimensional anisotropic models can also be resolved well except for some variations in the vertical component of the slowness that are not sensitive to the data. In 2-D isotropic models, “pure” lateral variations are not sensitive to the data, and when anisotropy is introduced, the result is that the horizontal and vertical component of the slowness cannot be estimated with the same spatial resolution because the null space is mostly related to horizontal and high frequency variations in the vertical component of the slowness. Since the distribution of singular values varies depending on the parametrization used, the effect of conventional regularization procedures in the final solution may also vary. When the model is isotropic, regularization translates into smoothness, and when the model is anisotropic regularization not only smooths but may also alter the anisotropy in the solution.

ROBUST REFERENCE-WATERMARKING SCHEME USING WAVELET PACKET TRANSFORM AND BIDIAGONAL-SINGULAR VALUE DECOMPOSITION

2009 ◽  
Vol 09 (03) ◽  
pp. 449-477 ◽  
Author(s):  
GAURAV BHATNAGAR ◽  
BALASUBRAMANIAN RAMAN
Keyword(s):  
Singular Value Decomposition ◽  
Gaussian Noise ◽  
Copyright Protection ◽  
Wavelet Packet ◽  
Singular Values ◽  
Wavelet Packet Transform ◽  
Singular Value ◽  
Noise Type ◽  
Value Decomposition ◽  
Watermarking Scheme

This paper presents a new robust reference watermarking scheme based on wavelet packet transform (WPT) and bidiagonal singular value decomposition (bSVD) for copyright protection and authenticity. A small gray scale logo is used as watermark instead of randomly generated Gaussian noise type watermark. A reference watermark is generated by original watermark and the process of embedding is done in wavelet packet domain by modifying the bidiagonal singular values. For the robustness and imperceptibly, watermark is embedded in the selected sub-bands, which are selected by taking into account the variance of the sub-bands, which serves as a measure of the watermark magnitude that could be imperceptibly embedded in each block. For this purpose, the variance is calculated in a small moving square window of size Sp× Sp(typically 3 × 3 or 5 × 5 window) centered at the pixel. A reliable watermark extraction is developed, in which the watermark bidiagonal singular values are extracted by considering the distortion caused by the attacks in neighboring bidiagonal singular values. Experimental evaluation demonstrates that the proposed scheme is able to withstand a variety of attacks and the superiority of the proposed method is carried out by the comparison which is made by us with the existing methods.

scholarly journals A Novel Wheelset Bearing Fault Diagnosis Method Integrated CEEMDAN, Periodic Segment Matrix, and SVD

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Chenguang Huang ◽  
Jianhui Lin ◽  
Jianming Ding ◽  
Yan Huang
Keyword(s):  
Fault Diagnosis ◽  
Singular Values ◽  
Ensemble Empirical Mode Decomposition ◽  
Singular Value ◽  
Bearing Fault Diagnosis ◽  
Mode Decomposition ◽  
Diagnosis Method ◽  
Adaptive Noise ◽  
Periodic Impulse ◽  
Value Decomposition

A novel fault diagnosis method, named CPS, is proposed based on the combination of CEEMDAN (complete ensemble empirical mode decomposition with adaptive noise), PSM (periodic segment matrix), and SVD (singular value decomposition). Firstly, the collected vibration signals are decomposed into a set of IMFs using CEEMDAN. Secondly, the PSM of the selected IMFs is constructed. Thirdly, singular values are obtained by SVD conducted on the space of PSM. Fourthly, the impulse components are enhanced by the singular value reconstruction with the first maximal singular value. Finally, the squared envelope spectra of the reconstructed signals are used to diagnose the wheelset bearing faults. The effectiveness of the proposed CPS has been verified by simulations and experiments. Compared to the well-known Hankel-based SVD, the proposed CPS performs better at extracting the weak periodic impulse responses from the measured signals with strong noise and interferences.

scholarly journals Hybrid Models Based on Singular Values and Autoregressive Methods for Multistep Ahead Forecasting of Traffic Accidents

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Lida Barba ◽  
Nibaldo Rodríguez
Keyword(s):  
Traffic Accidents ◽  
Singular Spectrum Analysis ◽  
Singular Values ◽  
Hybrid Models ◽  
Singular Value ◽  
Equivalent Models ◽  
Santiago De Chile ◽  
Artificial Neural Network Ann ◽  
Value Decomposition ◽  
Human Losses

The traffic accidents occurrence urges the intervention of researchers and society; the human losses and material damage could be abated with scientific studies focused on supporting prevention plans. In this paper prediction strategies based on singular values and autoregressive models are evaluated for multistep ahead traffic accidents forecasting. Three time series of injured people in traffic accidents collected in Santiago de Chile from 2000:1 to 2014:12 were used, which were previously classified by causes related to the behavior of drivers, passengers, or pedestrians and causes not related to the behavior as road deficiencies, mechanical failures, and undetermined causes. A simplified form of Singular Spectrum Analysis (SSA), combined with the autoregressive linear (AR) method, and a conventional Artificial Neural Network (ANN) are proposed. Additionally, equivalent models that combine Hankel Singular Value Decomposition (HSVD), AR, and ANN are evaluated. The comparative analysis shows that the hybrid models SSA-AR and SSA-ANN reach the highest accuracy with an averageMAPEof 1.5% and 1.9%, respectively, from 1- to 14-step ahead prediction. However, it was discovered that HSVD-AR shows a higher accuracy in the farthest horizons, from 12- to 14-step ahead prediction, which reaches an averageMAPEof 2.2%.

scholarly journals Simple Correspondence Analysis of Nominal-Ordinal Contingency Tables

2008 ◽  
Vol 2008 ◽  
pp. 1-17 ◽  
Author(s):  
Eric J. Beh
Keyword(s):  
Singular Value Decomposition ◽  
Correspondence Analysis ◽  
Contingency Table ◽  
Singular Values ◽  
Singular Value ◽  
Ordered Categories ◽  
Nominal Structure ◽  
Classical Technique ◽  
Versatile Tool ◽  
Value Decomposition

The correspondence analysis of a two-way contingency table is now accepted as a very versatile tool for helping users to understand the structure of the association in their data. In cases where the variables consist of ordered categories, there are a number of approaches that can be employed and these generally involve an adaptation of singular value decomposition. Over the last few years, an alternative decomposition method has been used for cases where the row and column variables of a two-way contingency table have an ordinal structure. A version of this approach is also available for a two-way table where one variable has a nominal structure and the other variable has an ordinal structure. However, such an approach does not take into consideration the presence of the nominal variable. This paper explores an approach to correspondence analysis using an amalgamation of singular value decomposition and bivariate moment decomposition. A benefit of this technique is that it combines the classical technique with the ordinal analysis by determining the structure of the variables in terms of singular values and location, dispersion and higher-order moments.

A fast algorithm for regularized focused 3D inversion of gravity data using randomized singular-value decomposition

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. G25-G34 ◽  
Author(s):  
Saeed Vatankhah ◽  
Rosemary Anne Renaut ◽  
Vahid Ebrahimzadeh Ardestani
Keyword(s):  
Linear System ◽  
Singular Value Decomposition ◽  
Fast Algorithm ◽  
Large Scale ◽  
Gravity Data ◽  
Singular Values ◽  
Singular Value ◽  
System Matrix ◽  
Large Scale System ◽  
Value Decomposition

We develop a fast algorithm for solving the under-determined 3D linear gravity inverse problem based on randomized singular-value decomposition (RSVD). The algorithm combines an iteratively reweighted approach for [Formula: see text]-norm regularization with the RSVD methodology in which the large-scale linear system at each iteration is replaced with a much smaller linear system. Although the optimal choice for the low-rank approximation of the system matrix with [Formula: see text] rows is [Formula: see text], acceptable results are achievable with [Formula: see text]. In contrast to the use of the iterative LSQR algorithm for the solution of linear systems at each iteration, the singular values generated using RSVD yield a good approximation of the dominant singular values of the large-scale system matrix. Thus, the regularization parameter found for the small system at each iteration is dependent on the dominant singular values of the large-scale system matrix and appropriately regularizes the dominant singular space of the large-scale problem. The results achieved are comparable with those obtained using the LSQR algorithm for solving each linear system, but they are obtained at a reduced computational cost. The method has been tested on synthetic models along with real gravity data from the Morro do Engenho complex in central Brazil.

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