Equivalent linearization for systems driven by Lévy white noise

The equivalent linearization method is extended to the case of nonlinear systems driven by Lévy white noise. The classical objective function used for the determination of the equivalent linear system cannot be applied because the Lévy process generating the Lévy white noise has no variance and higher order moments. An alternative objective function based on the characteristic function of the state of the linear system is used for solution. Two simple examples are presented to illustrate the proposed extension of the equivalent linearization method. The examples show that the proposed equivalent linearization method provides approximations of quality similar to the classical equivalent linearization method.     

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.     

Linear systems with polynomials of filtered Poisson processes

Moment equations are calculated exactly for the response of linear systems subjected polynomials of filtered Poisson processes. The Itô formula for stochastic differential equations driven by Poisson white noise is applied to derive moment equations. It is shown that the set of moment equations is closed. The proposed method is used to calculate moments up to the fourth order for the response of two linear systems subjected to quadratic forms of filtered Poisson processes. Results by Monte Carlo simulations are also presented for comparison.     

Non-Gaussian models for stochastic mechanics

Memoryless transformations of Gaussian processes and transformations with memory of the Brownian and Lévy processes are used to represent general non-Gaussian processes. The transformations with memory are solutions of stochastic differential equations driven by Gaussian and Lévy white noises. The processes obtained by these transformations are referred to as non-Gaussian models. Methods are developed for calibrating these models to records or partial probabilistic characteristics of non-Gaussian processes. The solution of the model calibration problem is not unique. There are different non-Gaussian models that are equivalent in the sense that they are consistent with the available information on a non-Gaussian process. The response analysis of linear and non-linear oscillators subjected to equivalent non-Gaussian models shows that some response statistics are sensitive to the particular equivalent non-Gaussian model used to represent the input. This observation is relevant for applications because the choice of a particular non-Gaussian input model can result in inaccurate predictions of system performance.     

The Itô and Stratonovich integrals for stochastic differential equations with poisson white noise

The relationship between the Itô and the Stratonovich integrals used for solving stochastic differential equations with Gaussian white noise is well known. However, this relationship seems to be less clear when dealing with stochastic differential equations driven by Poisson white noise. It is shown that there is no difference between the Itô and the Stratonovich integrals used to define the solution of stochastic differential equations with Poisson white noise. This result is in disagreement with findings of some previous publications but in agreement with the classical definition of the Itô and Stratonovich integrals. Intuitive considerations, arguments based on the theory of stochastic integrals with semimartingales, and examples are used to prove and demonstrate the claimed equality of the Itô and Stratonovich integrals.     

Multiplicative cases from additive cases: Extension of Kolmogorov–Feller equation to parametric Poisson white noise processes

In this paper the response of nonlinear systems driven by parametric Poissonian white noise is examined.As is well known, the response sample function or the response statistics of a system driven by external white noise processes is completely defined. Starting from the system driven by external white noise processes, when an invertible nonlinear transformation is applied, the transformed system in the new state variable is driven by a parametric type excitation. So this latter artificial system may be used as a tool to find out the proper solution to solve systems driven by parametric white noises. In fact, solving this new system, being the nonlinear transformation invertible, we must pass from the solution of the artificial system (driven by parametric noise) to that of the original one (driven by external noise, that is known). Moreover, introducing this invertible nonlinear transformation into the Itô’s rule for the original system driven by external input, one can derive the Itô’s rule for systems driven by a parametric type excitation, directly. In this latter case one can see how natural is the presence of the Wong–Zakai correction term or the presence of the hierarchy of correction terms in the case of normal and Poissonian white noise, respectively. Direct transformation on the Fokker–Planck and on the Kolmogorov–Feller equation for the case of parametric input are found.     

Stochastic response of linear and non-linear systems to α-stable Lévy white noises

The stochastic response of linear and non-linear systems to external α-stable Lévy white noises is investigated. In the literature, a differential equation in the characteristic function (CF) of the response has been recently derived for scalar systems only, within the theory of the so-called fractional Einstein–Smoluchowsky equations (FESEs). Herein, it is shown that the same equation may be built by rules of stochastic differential calculus, previously applied by one of the authors to systems driven by arbitrary delta-correlated processes. In this context, a straightforward formulation for multi-degree-of-freedom (MDOF) systems is also developed.Approximate CF solutions to the derived equation are sought for polynomial non-linearities, in stationary conditions. To this aim a wavelet representation is used, in conjunction with a weighted residual method. Numerical results prove in excellent agreement with exact solutions, when available, and digital simulation data.     

Equivalent linearization of MDOF systems under external Poisson white noise excitation

Equivalent linearization (EQL) techniques are developed and evaluated for multidimensional systems under external Poisson white noise excitation. Especially, a simulation strategy for the calculation of the linearization coefficients is proposed. The methods are illustrated by several examples that have been treated under Gaussian white noise excitation in the literature. It is shown that EQL for MDOF systems under Poisson white noise excitation is able to deal with problems of nearly the same dimension as under Gaussian white noise excitation.     

Feedback stabilization of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A procedure for designing a feedback control to asymptotic Lyapunov stability with probability one of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations is proposed. First, a one dimensional partially averaged Itô stochastic differential equation for controlled Hamiltonian is derived from the motion equations of the system by using the stochastic averaging method. Second, the dynamical programming equation for the ergodic control problem of the averaged system with undetermined cost function is set up based on the dynamical programming principle and the jump–diffusion chain stochastic differential rules. The optimal control law is obtained by solving the dynamical programming equation. Third, the analytical expression for the largest Lyapunov exponent of the averaged system is derived. Finally, the asymptotic Lyapunov stability with probability one of the originally controlled system is analyzed approximately by using the largest Lyapunov exponent. The cost function and optimal control forces are determined by the requirements of stabilizing the system. An example is worked out in detail to illustrate the effectiveness of the proposed method for stabilization control, and the control effect of the proposed feedback stabilization varies with the change of parameters is also studied in this paper, such as, the greater the excitation intensity of Gaussian and Poisson white noise, the better the stabilization control effect.     

Delay-independent stability of moments of a linear oscillator with delayed state feedback and parametric white noise

The stability of a linear oscillator with delayed state feedback driven by parametric Gaussian white noise is studied in this paper. The first and second order moment equations of the system response are derived by using moment method and Itô differential rule. Based on the moment equations, the delay-independent stable conditions of both moments are proposed: For the first order moment, the sufficient and necessary condition that guarantee delay-independent stability is identified to that of the deterministic system; for the second order moment, the sufficient condition that ensure delay-independent stability depends on noise intensity. The theoretical results are also illustrated with numerical simulations.     

Linear systems driven by martingale noise

A method is developed for calculating second moment properties and moments of order three and higher of the state X of a linear filter driven by martingale noise. The martingale noise is interpreted as the formal derivative of a square integrable martingale with continuous samples. The Gaussian white noise is an example of a martingale noise. It is shown that the differential equations of the mean and correlation functions of the state X developed in the paper resemble the corresponding equations of the classical linear random vibration and coincide with these equations if the input is a Gaussian white noise. The moment equations are derived by (1) the Itô formula for semimartingales and (2) the classical Itô formula applied to a diffusion process whose coordinates include X. An advantage of the second method is use of more familiar concepts. However, this method requires to calculate unnecessary moments and can be applied only for a class of martingale noise processes. Examples are presented to illustrate and evaluate the two methods for calculating moments of X and demonstrate the use of these methods in linear random vibration.     

The transient characteristics of an underdamped periodic potential system excited by two different kinds of noise

In this paper, the transient characteristics of an underdamped periodic potential system excited by multiplicative Gaussian white noise and additive Lévy noise are studied in terms of the mean first-passage time(MFPT) and the probability density function(PDF) of the first-passage time. The second-order underdamped periodic potential system is equivalent to two first-order stochastic differential equations. The MFPT was obtained by averaging the response value of the first-passage time, and the PDF image of the first-passage time was drawn under different parameter values. It was found that the increase in damping coefficient and stability index will inhibit the particle crossing, while the increase of multiplicative noise intensity, additive noise intensity and skewness parameter will promote the particle crossing to a certain extent.     

非高斯噪声激励下非线性漂移Fokker-Planck方程的非稳态解及其应用

非高斯噪声广泛存在于各种非线性系统，对非高斯噪声所驱动系统的非稳态演化行为进行研究可以更为深入的了解其内在的演化机理．本文对非高斯噪声和高斯白噪声共同驱动的非线性动力学系统的非稳态演化问题进行研究．首先应用格林函数的 $\Omega$ 展开理论在初始区域对非线性动力学系统进行线性化，然后结合本征值和本征矢理论推导出了该系统 Fokker-Planck 方程的近似非稳态解的表达式，最后以 Logistic 系统模型为例分析了非高斯噪声强度，关联时间及非高斯噪声偏离参数对非稳态解以及一阶矩的影响．研究结果表明，用 Logistic 模型描述产品产量增长时，其非稳态解可更好地反映产品产量在不稳定点附近的演化行为．     

Stochastic averaging of quasi-linear systems driven by Poisson white noise

The averaged generalized Itô and Fokker–Planck–Kolmogorov (FPK) equations for single-degree-of-freedom (SDOF) quasi-linear systems driven by Poisson white noise are derived and the approximate stationary solutions of the averaged generalized FPK equations are obtained by using the perturbation method for four typical quasi-linear systems, i.e., van der Pol oscillator, Rayleigh oscillator, system with energy-dependent damping, and system with power law damping. The effectiveness and accuracy of the perturbation solution are assessed by performing appropriate Monte Carlo simulations. It is found that analytical and numerical results agree well and the effect of non-Gaussianity of the excitation process is not negligible for predicting the probability densities of total energy and displacement of quasi-linear systems in most cases.     

Equations for probability density of response of dynamic systems to a class of non-Poisson random impulse process excitations

The excitation considered in the present paper is a random train of impulses driven by two classes of non-Poisson counting processes. The impulse processes are obtained by selecting impulses from a Poisson and from an Erlang-driven trains of impulses with the aid of an additional, purely jump stochastic process, assumed as an auxiliary state variable. The variable introduced for the first class of non-Poisson processes is governed by the stochastic differential equation driven by two independent Poisson processes, with different parameters, and is tantamount to a two-state Markov chain. The variable introduced for the second class of non-Poisson processes is governed by the stochastic differential equation driven by two independent Erlang processes, with different parameters. As each Erlang process is tantamount to a number of Markov states, the Markov chain for the whole problem is constructed. The equations governing the joint probability density-distribution function of the state vector of the dynamic system and of the Markov states are derived from the general integro-differential forward Chapman–Kolmogorov equation. The necessary jump probability intensity functions are evaluated for both classes of impulse processes and for purely external as well as parametric excitations. Parametric excitation multiplicative to the displacement and to the velocity state variable is considered. The resulting set of coupled integro-partial differential equations is obtained.     

Poisson white noise parametric input and response by using complex fractional moments

In this paper the solution of the generalization of the Kolmogorov–Feller equation to the case of parametric input is treated. The solution is obtained by using complex Mellin transform and complex fractional moments. Applying an invertible nonlinear transformation, it is possible to convert the original system into an artificial one driven by an external Poisson white noise process. Then, the problem of finding the evolution of the probability density function (PDF) for nonlinear systems driven by parametric non-normal white noise process may be addressed in determining the PDF evolution of a corresponding artificial system with external type of loading.     

Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method

In this paper the response in terms of probability density function of nonlinear systems under combined normal and Poisson white noise is considered. The problem is handled via a Path Integral Solution (PIS) that may be considered as a step-by-step solution technique in terms of probability density function. A nonlinear system under normal white noise, Poissonian white noise and under the superposition of normal and Poisson white noise is performed through PIS. The spectral counterpart of the PIS, ruling the evolution of the characteristic functions is also derived. It is shown that at the limit when the time step becomes an infinitesimal quantity an equation ruling the evolution of the probability density function of the response process of the nonlinear system in the presence of both normal and Poisson White Noise is provided.     

Invalidity of the spectral Fokker–Planck equation for Cauchy noise driven Langevin equation

The standard Langevin equation is a first order stochastic differential equation where the driving noise term is a Brownian motion. The marginal probability density is a solution to a linear partial differential equation called the Fokker–Planck equation. If the Brownian motion is replaced by so-called -stable noise (or Lévy noise) the Fokker–Planck equation no longer exists as a partial differential equation for the probability density because the property of finite variance is lost. Instead it has been attempted to formulate an equation for the characteristic function (the Fourier transform) corresponding to the density function. This equation is frequently called the spectral Fokker–Planck equation.

This paper raises doubt about the validity of the spectral Fokker–Planck equation in its standard formulation. The equation can be solved with respect to stationary solutions in the particular case where the noise is Cauchy noise and the drift function is a polynomial that allows the existence of a stationary probability density solution. The solution shows paradoxic properties by not being unique and only in particular cases having one of its solutions closely approximating the solutions to a corresponding Langevin difference equation. Similar doubt can be traced Grigoriu's work [Stochastic Calculus (2002)].