Power Spectral Density of Nonlinear System Response: The Recursion Method

1993 ◽  
Vol 60 (2) ◽  
pp. 358-365 ◽  
Author(s):  
R. Vale´ry Roy ◽  
P. D. Spanos
Keyword(s):  
Broad Class ◽  
Formal Solution ◽  
Power Series Expansion ◽  
Kolmogorov Equation ◽  
System Response ◽  
Lanczos Algorithm ◽  
Spectral Densities ◽  
Recursion Method ◽  
Approximate Form ◽  
Single Well

Spectral densities of the response of nonlinear systems to white noise excitation are considered. By using a formal solution of the associated Fokker-Planck-Kolmogorov equation, response spectral densities are represented by formal power series expansion for large frequencies. The coefficients of the series, known as the spectral moments, are determined in terms of first-order response statistics. Alternatively, a J-fraction representation of spectral densities can be achieved by using a generalization of the Lanczos algorithm for matrix tridiagonalization, known as the “recursion method.” Sequences of rational approximations of increasing order are obtained. They are used for numerical calculations regarding the single-well and double-well Duffing oscillators, and Van der Pol type oscillators. Digital simulations demonstrate that the proposed approach can be quite reliable over large variations of the system parameters. Further, it is quite versatile as it can be used for the determination of the spectrum of the response of a broad class of randomly excited nonlinear oscillators, with the sole prerequisite being the availability, in exact or approximate form, of the stationary probability density of the response.

Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements

2016 ◽  
Vol 83 (12) ◽  
Author(s):  
Pol D. Spanos ◽  
Alberto Di Matteo ◽  
Yezeng Cheng ◽  
Antonina Pirrotta ◽  
Jie Li
Keyword(s):  
Fractional Derivative ◽  
Survival Probability ◽  
Stochastic Averaging ◽  
Kolmogorov Equation ◽  
System Response ◽  
Statistical Linearization ◽  
Van Der Pol ◽  
Galerkin Scheme ◽  
Confluent Hypergeometric Functions

In this paper, an approximate semi-analytical approach is developed for determining the first-passage probability of randomly excited linear and lightly nonlinear oscillators endowed with fractional derivative elements. The amplitude of the system response is modeled as one-dimensional Markovian process by employing a combination of the stochastic averaging and the statistical linearization techniques. This leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. Next, an approximate solution of this equation is sought by resorting to a Galerkin scheme. Specifically, a convenient set of confluent hypergeometric functions, related to the corresponding linear oscillator with integer-order derivatives, is used as orthogonal basis for this scheme. Applications to the standard viscous linear and to nonlinear (Van der Pol and Duffing) oscillators are presented. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed approximate analytical solution.

STEADY-STATE DISTRIBUTIONS FOR HARMONIC OSCILLATOR WITH VERY FAST FREQUENCY FLUCTUATIONS

2012 ◽  
Vol 11 (03) ◽  
pp. 1242009 ◽  
Author(s):  
ALEXANDER DUBKOV
Keyword(s):  
Steady State ◽  
Harmonic Oscillator ◽  
Probability Density ◽  
Joint Probability ◽  
Probability Density Functions ◽  
Kolmogorov Equation ◽  
Approximate Form ◽  
Frequency Fluctuations ◽  
The Moment ◽  
Joint Probability Density

The moment and probability steady-state characteristics of harmonic oscillator with frequency fluctuations in the form of white noise are investigated. Based on well-known functional approach, we derive integro-differential Kolmogorov equation for the joint probability density function of oscillator coordinate and velocity. For white Gaussian noise, using a set of equations for joint moments, we reconstruct the approximate form of coordinate and velocity distributions in the limit of small friction. As shown, these probability density functions do not exist for zero friction because they cannot be normalized.

Nonstationary Response of Optimal Controlled Stochastic Van Der Pol Oscillator

2014 ◽  
Vol 875-877 ◽  
pp. 2000-2005
Author(s):  
Lu Yuan Qi ◽  
Wei Xu ◽  
Wei Ting Gao
Keyword(s):  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Van Der Pol Oscillator ◽  
Kolmogorov Equation ◽  
System Response ◽  
Control Law ◽  
Dynamical Programming ◽  
Control Constraint ◽  
Van Der Pol ◽  
Ito Equation

A procedure to calculate the transient response of optimal controlled stochastic Van Der Pol oscillator is proposed. The stochastic averaging method is employed to obtain a partially averaged Itô equation for the amplitude process. The dynamical programming equation is adopted to minimize the system response. An optimal control law with a control constraint is established. The completed averaged Itô equation is obtained. The transient probability density function is solved from Fokker-Planck-Kolmogorov equation by Galerkin method. Results obtained show the proposed method is accurate. The effective of the control strategy is significant.

scholarly journals Näherungslösungen für die Form kernmagnetischer Signale bei Impulsexperimenten in Festkörpern

1968 ◽  
Vol 23 (8) ◽  
pp. 1194-1201
Author(s):  
G. Siegle
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Phase Difference ◽  
Pulse Sequence ◽  
Formal Solution ◽  
Power Series Expansion ◽  
Maximum Amplitude ◽  
Experimental Results ◽  
Longitudinal Relaxation ◽  
Pulse Distance

NMR transients in solids have been calculated using the method of LOWE and NORBERG 1 and its extension by POWLES and STRANGE 2. Some remarks are given concerning the evaluation of the formal solution for the signal after several pulses.So far the influence of the pulse durations and of the longitudinal relaxation is neglected usually, higher terms of the power series expansion of the signals were discussed only with respect to corrections of their amplitudes. Some remarkable differences between these calculations and new experimental results to be reported here are explained — at least in principle — avoiding these limitations:The maximum amplitude of the echo after two 90°-pulses (phase difference of their modulations Δφ=90°) occurs not exactly at a time twice the pulse distance, the deviation is explained considering the duration of the pulses and the decay of the echo envelope.After two pulses with Δφ=0° an echo was found which should disappear only in an ideal twospin system. For a three pulse sequence the influence of the longitudinal relaxation is described; and in a multi-pulse sequence the calculated modulation of the amplitude of successive echoes was checked by experiments.

scholarly journals Fast Estimation of Approximate Matrix Ranks Using Spectral Densities

2017 ◽  
Vol 29 (5) ◽  
pp. 1317-1351 ◽  
Author(s):  
Shashanka Ubaru ◽  
Yousef Saad ◽  
Abd-Krim Seghouane
Keyword(s):  
Spectral Density ◽  
Large Data ◽  
Lanczos Algorithm ◽  
Spectral Densities ◽  
Matrix Rank ◽  
Density Distributions ◽  
Fast Estimation ◽  
The Matrix ◽  
Rank One ◽  
Probability Density Distributions

Many machine learning and data-related applications require the knowledge of approximate ranks of large data matrices at hand. This letter presents two computationally inexpensive techniques to estimate the approximate ranks of such matrices. These techniques exploit approximate spectral densities, popular in physics, which are probability density distributions that measure the likelihood of finding eigenvalues of the matrix at a given point on the real line. Integrating the spectral density over an interval gives the eigenvalue count of the matrix in that interval. Therefore, the rank can be approximated by integrating the spectral density over a carefully selected interval. Two different approaches are discussed to estimate the approximate rank, one based on Chebyshev polynomials and the other based on the Lanczos algorithm. In order to obtain the appropriate interval, it is necessary to locate a gap between the eigenvalues that correspond to noise and the relevant eigenvalues that contribute to the matrix rank. A method for locating this gap and selecting the interval of integration is proposed based on the plot of the spectral density. Numerical experiments illustrate the performance of these techniques on matrices from typical applications.

A Study of the Spectral Functions and Complex Band Structures for the Disordered Binary Alloys: AgPd and AuFe

1997 ◽  
Vol 11 (31) ◽  
pp. 3703-3713 ◽  
Author(s):  
Parthapratim Biswas ◽  
Biplab Sanyal ◽  
A. Mookerjee ◽  
Nasreen Chowdhury ◽  
Mesbahuddin Ahmed ◽  
...  
Keyword(s):  
Binary Alloys ◽  
Second Order ◽  
Spectral Densities ◽  
Spectral Functions ◽  
Band Structures ◽  
Recursion Method ◽  
Augmented Space ◽  
Alloy Systems

We apply the augmented space recursion method to the study of the spectral densities and the complex band structures of two alloy systems AgPd and AuFe, both in the first and second order TB-LMTO approximations.

scholarly journals Singular Perturbation of Nonlinear Systems with Regular Singularity

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Domingos H. U. Marchetti ◽  
William R. P. Conti
Keyword(s):  
Nonlinear Systems ◽  
Power Series ◽  
Bessel Functions ◽  
Formal Solution ◽  
Nonlinear Partial Differential Equations ◽  
Power Series Expansion ◽  
Singularly Perturbed ◽  
Type Condition ◽  
Regular Singularity ◽  
Evolution Type

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form εzf′=Fε,z,f with F a Cν-valued function, holomorphic in a polydisc D-ρ×D-ρ×D-ρν. We show that its unique formal solution in power series of ε, whose coefficients are holomorphic functions of z, is 1-summable under a Siegel-type condition on the eigenvalues of Ff(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type.

Numerical Simulation and Prediction of High Fluorine Groundwater Transport in Zhangye Basin

2012 ◽  
Vol 466-467 ◽  
pp. 36-41
Author(s):  
Yong Hui An ◽  
Shuang Bao Han ◽  
Xi Wu ◽  
Xu Xue Cheng ◽  
Wei Po Liu
Keyword(s):  
Transport Model ◽  
System Response ◽  
Confined Water ◽  
Local Planning ◽  
Deep Groundwater ◽  
Well Completion ◽  
Phreatic Water ◽  
Single Well ◽  
Zhangye Basin ◽  
Exploitation Intensity

Simulate Feflow with finite element method, and establish flow model and solute transport model of high fluorine groundwater area in Zhangye Basin. Predicting groundwater system response under different exploitation scheme, and evaluating the risk of deep low fluorine groundwater polluted by shallow high fluorine groundwater. The results showed that, firstly, the existing exploitation intensity and the increasing exploitation intensity of the local planning would lead to groundwater table descent, for the deep groundwater head is higher than phreatic water in above scheme, the polluted risk of low fluorine freshwater is low. Secondly, low and deep groundwater “cross strata” caused by well completion technology would lead to deep low fluorine freshwater polluted, and the high fluorine polluted area caused by single well is limited, but the polluted risk of low fluorine aquifer is increasing with that confined water head is lower than phreatic water in local concentrated excess exploitation area. Propose the measures and suggestions of the groundwater sustainable utilization.

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