Random vibration of hysteretic systems under Poisson white noise excitations

Hysteresis widely exists in civil structures,and dissipates the mechanical energy of systems.Research on the random vibration of hysteretic systems,however,is still insufficient,particularly when the excitation is non-Gaussian.In this paper,the radial basis function(RBF)neural network(RBF-NN)method is adopted as a numerical method to investigate the random vibration of the Bouc-Wen hysteretic system under the Poisson white noise excitations.The solution to the reduced generalized Fokker-PlanckKolmogorov(GFPK)equation is expressed in terms of the RBF-NNs with the Gaussian activation functions,whose weights are determined by minimizing the loss function of the reduced GFPK equation residual and constraint associated with the normalization condition.A steel fiber reinforced ceramsite concrete(SFRCC)column loaded by the Poisson white noise is studied as an example to illustrate the solution process.The effects of several important parameters of both the system and the excitation on the stochastic response are evaluated,and the obtained results are compared with those obtained by the Monte Carlo simulations(MCSs).The numerical results show that the RBF-NN method can accurately predict the stationary response with a considerable high computational efficiency.

Analytical Fitting Functions of Finite Sample Discrete Entropies of White Gaussian Noise

In order to find the convergence rate of finite sample discrete entropies of a white Gaussian noise(WGN), Brown entropy algorithm is numerically tested.With the increase of sample size, the curves of these finite sample discrete entropies are asymptotically close to their theoretical values.The confidence intervals of the sample Brown entropy are narrower than those of the sample discrete entropy calculated from its differential entropy, which is valid only in the case of a small sample size of WGN. The differences between sample Brown entropies and their theoretical values are fitted by two rational functions exactly, and the revised Brown entropies are more efficient. The application to the prediction of wind speed indicates that the variances of resampled time series increase almost exponentially with the increase of resampling period.

SELF-TUNING WEIGHTED MEASUREMENT FUSION WHITE NOISE DECONVOLUTION ESTIMATOR

For the multi-sensor linear discrete time-invariant stochastic systems with correlated measurement noises and unknown noise statistics,an on-line noise statistics estimator is obtained using the correlation method.Substituting it into the optimal weighted fusion steady-state white noise deconvolution estimator based on the Kalman filtering,a self-tuning weighted measurement fusion white noise deconvolution estimator is presented.By the Dynamic Error System Analysis(DESA) method,it proved that the self-tuning fusion white noise deconvolution estimator converges to the steady-state optimal fusion white noise deconvolution estimator in a realization.Therefore,it has the asymptotically global optimality.A simulation example for the tracking system with 3 sensors and the Bernoulli-Gaussian input white noise shows its effectiveness.

THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE

This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation ut+(-Δ)^(s1)ut+β(-Δ)^(s2)u=F(u,x,t)subject o random Gaussian white noise for initial and final data.Under the suitable assumptions s1,s2andβ,we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard,which are mainly driven by random noise.Moreover,we propose the Fourier truncation method for stabilizing the above ill-posed problems.We derive an error estimate between the exact solution and its regularized solution in an E‖·‖Hs22norm,and give some numerical examples illustrating the effect of above method.

Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation

The nonstationary probability densities of system response of a single-degree- of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied. Using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokl^er-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Then, the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained. A van der Pol-Duffing oscillator subject to modulated white noise is given as an example to illustrate the proposed procedures. The effects of the system parameters, such as the linear damping coefficient and the nonlinear stiffness coefficient, on the system response are discussed.

Dynamics of a prey-predator system under Poisson white noise excitation

The classical Lotka-Volterra （LV） model is a well-known mathematical model for prey-predator ecosystems. In the present paper, the pulse-type version of stochastic LV model, in which the effect of a random natural environment has been modeled as Poisson white noise, is in- vestigated by using the stochastic averaging method. The averaged generalized It6 stochastic differential equation and Fokkerlanck-Kolmogorov （FPK） equation are derived for prey-predator ecosystem driven by Poisson white noise. Approximate stationary solution for the averaged generalized FPK equation is obtained by using the perturbation method. The effect of prey self-competition parameter e2s on ecosystem behavior is evaluated. The analytical result is confirmed by corresponding Monte Carlo （MC） simulation.

Random Attractor Family for the Kirchhoff Equation of Higher Order with White Noise

The existence of random attractor family for a class of nonlinear high-order Kirchhoff equation stochastic dynamical systems with white noise is studied. The Ornstein-Uhlenbeck process and the weak solution of the equation are used to deal with the stochastic terms. The equation is transformed into a general stochastic equation. The bounded stochastic absorption set is obtained by estimating the solution of the equation and the existence of the random attractor family is obtained by isomorphic mapping method. Temper random compact sets of random attractor family are obtained.

A Random Attractor Family of the High Order Beam Equations with White Noise

In this paper, we studied a class of damped high order Beam equation stochas-tic dynamical systems with white noise. First, the Ornstein-Uhlenbeck process is used to transform the equation into a noiseless random equation with random variables as parameters. Secondly, by estimating the solution of the equation, we can obtain the bounded random absorption set. Finally, the isomorphism mapping method and compact embedding theorem are used to obtain the system. It is progressively compact, then we can prove the existence of ran-dom attractors.

EXACT SOLUTIONS FOR STATIONARY RESPONSES OF SEVERAL CLASSES OF NONLINEAR SYSTEMS TO PARAMETRIC AND/OR EXTERNAL WHITE NOISE EXCITATIONS

The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and. as an example, the equivalent stochastic systems for the second order G.E. D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may he found by searching the equivalent G.E.D. systems.

ON THE TWO BIFURCATIONS OF A WHITE-NOISE EXCITED HOPF BIFURCATION SYSTEM

The present work is concerned with the behavior of the second bifurcation of a Hopf bifurcation system excited by white-noise. It is found that the intervention of noises induces a drift of the bifurcation point along with the subtantial change in bifurcation type.

The Equilibrium Range of Wind Wave Spectra: an Explanation Based on White Noise

Laboratory experiments and field observations show that the equilibrium range of wind wave spectra presents a – 4 power law when it is scaled properly. This feature has been attributed to energy balance in spectral space by many researchers. In this paper we point out that white noise on an oscillation system can also lead to a similar inverse power law in the corresponding displacement spectrum, implying that the – 4 power law for the equilibrium range of wind wave spectra may probably only reflect the randomicity of the wind waves rather than any other dynamical processes in physical space. This explanation may shed light on the mechanism of other physical processes with spectra also showing an inverse power law, such as isotropic turbulence, internal waves, etc.

The fractional coupled KdV equations: Exact solutions and white noise functional approach

Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.

The Stochastic stability of a Logistic model with Poisson white noise

The stochastic stability of a logistic model subjected to the effect of a random natural environment, modeled as Poisson white noise process, is investigated. The properties of the stochastic response are discussed for calculating the Lyapunov exponent, which had proven to be the most useful diagnostic tool for the stability of dynamical systems. The generalised Itp differentiation formula is used to analyse the stochastic stability of the response. The results indicate that the stability of the response is related to the intensity and amplitude distribution of the environment noise and the growth rate of the species.

A MODEL FOR WHITE NOISE ANALYSIS IN P-ADIC NUMBER FIELDS

We propose a simple model for white noise analysis in the p-adic case based on the p-adic analogue of the space L(2)(R(infinity), d mu).

GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION

A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of Koopman operators for chaotic dynamical systems, in terms of annihilation operators partial derivative(t) and its dual, creation operators partial derivative(t)*.

Stochastic Response Analysis of Piled Offshore Platform Excited by Stationary Filtered White Noise

In this paper, the analysis method of stochastic response of piled offshore platform excited by stationary filtered white noise is presented. With this method, the strong ground motion is considered as three direction stationary filtered white noise process, the theoretic solutions of three special integration equations are derived with the residue theorem, and the expression of response nodal displacements and member forces of offshore platform excited by the stationary filtered white noise is put forward. The stochastic response of a piled offshore platform excited by the stationary filtered white noise, which is located 114.3 m in water depth, is computed. The results are compared with those obtained with the response spectrum analysis method and the stationary white noise model analysis method, and the corresponding conclusion is drawn.

GAUSS WHITE NOISE PERTURBATIONS OF NONHOLONOMIC MECHANICAL SYSTEMS

The perturbations of nonholonomic mechanical systems under the Gauss white noises are studied in this paper. It is proved that the differential equations of the first-order moments of the solution process coincide with the corresponding equations in the non-perturbational case, and that there are e2 -terms but no e-terms in the differential equations of the second-order moments. Two propositions are obtained. Finally, an example is given to illustrate the application of the results.

Self-tuning measurement fusion white noise deconvolution estimator with correlated noises

For the multisensor linear discrete time-invariant stochastic systems with correlated noises and unknown noise statistics,an on-line noise statistics estimator is presented by using the correlation method.Substituting it into the steady-state Riccati equation,the self-tuning Riccati equation is obtained.Using the Kalman filtering method,based on the self-tuning Riccati equation,a self-tuning weighted measurement fusion white noise deconvolution estimator is presented.By the dynamic error system analysis（DESA） method,it is proved that the self-tuning fusion white noise deconvolution estimator converges to the optimal fusion steadystate white noise deconvolution estimator in a realization,so that it has the asymptotic global optimality.A simulation example for Bernoulli-Gaussian input white noise shows its effectiveness.

Some Applications of Higher Moments of the Linear Gaussian White Noise Process

The Linear Gaussian white noise process is an independent and identically distributed (iid) sequence with zero mean and finite variance with distribution N (0, σ2 ) . Hence, if X1, x2, …, Xn is a realization of such an iid sequence, this paper studies in detail the covariance structure of X1d, X2d, …, Xnd, d=1, 2, …. By this study, it is shown that: 1) all powers of a Linear Gaussian White Noise Process are iid but, not normally distributed and 2) the higher moments (variance and kurtosis) of Xtd, d=2, 3, … can be used to distinguish between the Linear Gaussian white noise process and other processes with similar covariance structure.

Stochastic stability of FitzHugh-Nagumo systems perturbed by Gaussian white noise

The current paper is devoted to the study of the stochastic stability of FitzHugh-Nagumo systems perturbed by Gaussian white noise. First, the dynamics of stochastic FitzHugh-Nagumo systems are studied. Then, the existence and uniqueness of their invariant measures, which mix exponentially are proved. Finally, the asymptotic behaviors of invariant measures when size of noise gets to zero are investigated.