Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II

1988 ◽  
Vol 55 (3) ◽  
pp. 702-705 ◽  
Author(s):  
Y. K. Lin ◽  
Guoqiang Cai
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Probability Distribution ◽  
White Noise ◽  
Probability Density ◽  
Detailed Balance ◽  
Stationary Probability ◽  
Probability Flow ◽  
Parametric And External Excitations ◽  
White Noises

A systematic procedure is developed to obtain the stationary probability density for the response of a nonlinear system under parametric and external excitations of Gaussian white noises. The procedure is devised by separating the circulatory portion of the probability flow from the noncirculatory flow, thus obtaining two sets of equations that must be satisfied by the probability potential. It is shown that these equations are identical to two of the conditions established previously under the assumption of detailed balance; therefore, one remaining condition for detailed balance is superfluous. Three examples are given for illustration, one of which is capable of exhibiting limit cycle and bifurcation behaviors, while another is selected to show that two different systems under two differents sets of excitations may result in the same probability distribution for their responses.

Exact Stationary-Response Solution for Second Order Nonlinear Systems Under Parametric and External White-Noise Excitations

1987 ◽  
Vol 54 (2) ◽  
pp. 414-418 ◽  
Author(s):  
Y. Yong ◽  
Y. K. Lin
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Gaussian White Noise ◽  
Detailed Balance ◽  
Stationary Solutions ◽  
Second Order ◽  
Unexpected Result ◽  
Sporadic Cases ◽  
Suitable Combination ◽  
Parametric And External Excitations

Except for rare sporadic cases, exact stationary solutions for second order nonlinear systems under Gaussian white-noise excitations are known only for certain types of systems and only when the excitations are purely external (additive). Yet, in many engineering problems, random excitations may also be parametric (multiplicative). It is shown in this paper that the method of detailed balance developed by the physicists can be applied to obtain systematically the stationary solutions for a large class of nonlinear systems under either external random excitations or parametric random excitations, or both. Examples are given for those cases where solutions have been given previously in the literature as well as other cases where solutions are new. An unexpected result is revealed in one of the new solutions, namely, under suitable combination of the parametric and external excitations of Gaussian white noises, the response of a nonlinear system can be Gaussian.

Equivalent Nonlinear System Method for Stochastically Excited Hamiltonian Systems

1994 ◽  
Vol 61 (3) ◽  
pp. 618-623 ◽  
Author(s):  
W. Q. Zhu ◽  
T. T. Soong ◽  
Y. Lei
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
White Noise ◽  
Probability Density ◽  
Hamiltonian Systems ◽  
Gaussian White Noise ◽  
Energy Balancing ◽  
Dissipation Energy ◽  
System Method ◽  
Approximate Probability

An equivalent nonlinear system method is presented to obtain the approximate probability density for the stationary response of multi-degree-of-freedom nonlinear Hamiltonian systems to Gaussian white noise parametric and/or external excitations. The equivalent nonlinear systems are obtained on the basis of one of the following three criteria: least mean-squared deficiency of damping forces, dissipation energy balancing, and least mean-squared deficiency of dissipation energies. An example is given to illustrate the application and validity of the method and the differences in the three equivalence criteria.

scholarly journals Dynamics of a Nonlinear Business Cycle Model Under Poisson White Noise Excitation

2015 ◽  
Vol 3 (2) ◽  
pp. 176-183 ◽  
Author(s):  
Jiaorui Li ◽  
Shuang Li
Keyword(s):  
Business Cycle ◽  
White Noise ◽  
Economic System ◽  
Probability Density ◽  
Stationary Probability ◽  
Cycle Model ◽  
Poisson White Noise ◽  
Stationary Probability Density ◽  
White Noise Excitation ◽  
Business Cycle Model

AbstractSeveral observations in real economic systems have shown the evidence of non-Gaussianity behavior, and one of mathematical models to describe these behaviors is Poisson noise. In this paper, stationary probability density of a nonlinear business cycle model under Poisson white noise excitation has been studied analytically. By using the stochastic averaged method, the approximate stationary probability density of the averaged generalized FPK equations are obtained analytically. The results show that the economic system occurs jump and bifurcation when there is a Poisson impulse existing in the periodic economic system. Furthermore, the numerical solutions are presented to show the effectiveness of the obtained analytical solutions.

scholarly journals Analysis of exceedance probability of displacement response of randomly nonlinear structures

2000 ◽  
Vol 22 (4) ◽  
pp. 212-224 ◽  
Author(s):  
Luu Xuan Hung
Keyword(s):  
Probability Density Function ◽  
Nonlinear Systems ◽  
White Noise ◽  
Probability Density ◽  
Density Function ◽  
Exceedance Probability ◽  
Nonlinear Structures ◽  
Exact Probability ◽  
Displacement Response ◽  
Linearization Techniques

The paper presents the estimation of the exact exceedance probability (EEP) of stationary responses of some white noise-randomly excited nonlinear systems whose exact probability density function can be known. Consequently, the approximate exceedance probabilities (AEPs) are evaluated based on the analysis of equivalent linearized systems using the traditional Caughey method and the extension technique of LOMSEC. Comparisons of the AEPs versus the EEP are demonstrated. The obtained results indicate important characters of the exceedance probability (EP) as well as the influence of nonlinearity over EP. The evaluation of the applied possibility of the proposed linearization techniques for estimating AEPs are made.

Path Integral Method for Nonlinear Systems Under Levy White Noise

2017 ◽  
Vol 3 (3) ◽  
Author(s):  
Alberto Di Matteo ◽  
Antonina Pirrotta
Keyword(s):  
Probability Density Function ◽  
Nonlinear Systems ◽  
Probability Density ◽  
Density Function ◽  
Path Integral ◽  
Response Probability ◽  
Stability Index ◽  
Integral Solution ◽  
Stable Random Variables ◽  
White Noises

In this paper, the probabilistic response of nonlinear systems driven by alpha-stable Lévy white noises is considered. The path integral solution is adopted for determining the evolution of the probability density function of nonlinear oscillators. Specifically, based on the properties of alpha-stable random variables and processes, the path integral solution is extended to deal with Lévy white noises input with any value of the stability index alpha. It is shown that at the limit when the time increments tend to zero, the Einstein–Smoluchowsky equation, governing the evolution of the response probability density function, is fully restored. Application to linear and nonlinear systems under different values of alpha is reported. Comparisons with pertinent Monte Carlo simulation data and analytical solutions (when available) demonstrate the accuracy of the results.

scholarly journals Complex Dynamics of Credit Risk Contagion with Time-Delay and Correlated Noises

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Tingqiang Chen ◽  
Xindan Li ◽  
Jianmin He
Keyword(s):  
Dynamical System ◽  
Time Delay ◽  
Probability Distribution ◽  
Credit Risk ◽  
Resistance Coefficient ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Credit Risk Contagion ◽  
Nonlinear Resistance ◽  
White Noises

The stochastic time-delayed system of credit risk contagion driven by correlated Gaussian white noises is investigated. Novikov’s theorem, the time-delay approximation, the path-integral approach, and first-order perturbation theory are used to derive time-delayed Fokker-Planck model and the stationary probability distribution function of the dynamical system of credit risk contagion in the financial market. Using the method of numerical simulation, the Hopf bifurcation and chaotic behaviors of credit risk contagion are analyzed when time-delay and nonlinear resistance coefficient are varied and the effects of time-delay, nonlinear resistance and the intensity and the correlated degree of correlated Gaussian white noises on the stationary probability distribution of credit risk contagion are investigated. It is found that, as the infectious scale of credit risk and the wavy frequency of credit risk contagion are increased, the stability of the system of credit risk contagion is reduced, the dynamical system of credit risk contagion gives rise to chaotic phenomena, and the chaotic area increases gradually with the increase in time-delay. The nonlinear resistance only influences the infectious scale and range of credit risk, which is reduced when the nonlinear resistance coefficient increases. In addition, the curve of the stationary probability distribution is monotone decreasing with the increase in parameters value of time-delay, nonlinear resistance, and the intensity and the correlated degree of correlated Gaussian white noises.

WHITE NOISE PATH INTEGRAL TREATMENT OF THE PROBABILITY DISTRIBUTION FOR THE AREA ENCLOSED BY A POLYMER LOOP IN CROSSED ELECTRIC-MAGNETIC FIELDS

2012 ◽  
Vol 17 ◽  
pp. 77-82
Author(s):  
BEVERLY V. GEMAO ◽  
JINKY B. BORNALES
Keyword(s):  
Magnetic Field ◽  
Brownian Motion ◽  
Probability Distribution ◽  
Magnetic Fields ◽  
White Noise ◽  
Probability Density ◽  
Path Integral ◽  
Integral Treatment ◽  
The Many ◽  
The Magnetic Field

The probability density for the area A enclosed by a polymer loop in crossed electric-magnetic fields is evaluated using the Hida-Streit formulation. In this approach, the many possible conformations of the polymer, x(v) and y(v), are represented by paths and are parametrized in terms Brownian motion. When the magnetic field is switched off, results agree with the works of Khandekar and Wiegel5

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