Transient Probability Density Functions Research Articles

A full‐probabilistic cloud analysis for structural seismic fragility via decoupled M‐PDEM

AbstractPerformance‐based earthquake engineering (PBEE) is essential for ensuring engineering safety. Conducting seismic fragility analysis within this framework is imperative. Existing methods for seismic fragility analysis often rely heavily on double loop reanalysis and empirical data fitting, leading to challenges in obtaining high‐precision results with a limited number of representative structural analysis instances. In this context, a new methodology for seismic fragility based on a full‐probabilistic cloud analysis is proposed via the decoupled multi‐probability density evolution method (M‐PDEM). In the proposed method, the assumption of a log‐normal distribution is not required. According to the random event description of the principle of preservation of probability, the transient probability density functions (PDFs) of intensity measure (IM) and engineering demand parameter (EDP), as key response quantities of the seismic‐structural system, are governed by one‐dimensional Li‐Chen equations, where the physics‐driven forces are determined by representative analysis data of the stochastic dynamic system. By generating ground motions based on representative points of basic random variables and performing structural dynamic analysis, the decoupled M‐PDEM is employed to solve the one‐dimensional Li‐Chen equations. This yields the joint PDF of IM and EDP, as well as the conditional PDF of EDP given IM, resulting in seismic fragility analysis outcomes. The numerical implementation procedure is elaborated in detail, and validation is performed using a six‐story nonlinear reinforced concrete (RC) frame subjected to non‐stationary stochastic ground motions. Comparative analysis against Monte Carlo simulation (MCS) and traditional cloud analysis based on least squares regression (LSR) reveals that the proposed method achieves higher computational precision at comparable structural analysis costs. By directly solving the physics‐driven Li‐Chen equations, the method provides the full‐probabilistic joint information of IM and EDP required for cloud analysis, surpassing the accuracy achieved by traditional methods based on statistical moment fitting and empirical distribution assumptions.

Transient PDF Solution of Nonlinear Stochastic Oscillator Subjected to Modulated Gaussian White Noise

In this paper, an extended exponential-polynomial-closure (EPC) procedure is presented to investigate the transient probability density function (PDF) of stochastic oscillator under the stimulation being modulated white noise based on the conventional EPC approach. By the presented EPC procedure, an evolutionary exponential–polynomial function is defined as the PDF solution and a set of evolutionary independent base functions are introduced to span a [Formula: see text] space. By integrating the residual error with those evolutionary functions spanning the [Formula: see text] space, a group of ordinary differential equations can then be formulated. Three nonlinear oscillators are analyzed numerically and the results comparisons of various techniques validate the presented EPC procedure in solution efficiency and accuracy, particularly in the PDF solution tails which play a crucial role in reliability analysis. The transient PDF solutions acquired by the presented EPC procedure match well with Monte Carlo simulation even in the PDF solution tails, whereas the solutions obtained by the equivalent linearization are deemed to be unacceptable for the strongly nonlinear oscillators subjected to modulated stimulation. Notably, the presented EPC procedure can highly increase the computational efficiency by about 1[Formula: see text]000 times in all the presented numerical examples in comparison to Monte Carlo simulation.

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

Stochastic bifurcations and transient dynamics of probability responses with radial basis function neural networks

Stochastic noise is common in many engineering systems. Stochastic noise and nonlinearity of dynamical systems will lead to the occurrence of complex dynamic phenomena, such as stochastic bifurcations and transitions. In this paper, a radial basis function neural networks (RBF-NN) method is applied to study stochastic bifurcations and transient dynamics of probability response. A Duffing oscillator under additive and multiplicative random noise is presented to show the effectiveness of the proposed solution method. A new type of stochastic bifurcations is found with an increase of system control parameter, which is called a stochastic double P-bifurcation. It occurs when a stationary probability density function (PDF) changes from a single peak to another single one through two double peaks with their maximum exchange. It is revealed that such a double P-bifurcation is linked to two catastrophic bifurcations of its deterministic counterpart. Moreover, these different types of transient evolutions of the PDFs are observed with transitions of probability maximum peak to one, and to the other, as well as to the both of stable invariant sets from different initial PDFs. The evolutionary orientation of transient PDFs aligns with unstable invariant manifolds leading to stable invariant sets. The previous conjecture is confirmed that the noise-induced transient dynamics and bifurcations are dominated by the global topology and its sudden changes of corresponding deterministic systems without noise.

Application of Complex Fractional Moment in nonlinear system with Gaussian colored noise

This paper studies the transient response of the nonlinear system with Gaussian colored noise. The one-dimensional Itô stochastic differential equation with respect to amplitude response is obtained by the stochastic averaging technique, and the Mellin transformation is applied to the related Fokker–Planck–Kolmogorov (FPK) equation to obtain a set of Complex Fractional Moment (CFM) differential equations. The semi-analytical solution of the FPK equation can be reconstructed by solving the CFM equations. Numerical method is adopted to verify the validity of the CFM method, and the dynamical temporal behaviors of the transient response Probability Density Function (PDF) are analyzed. In addition, the transient bifurcation phenomenon of the system is discussed, and the 3-D distribution diagram of peak value of transient PDF is produced.

Transient response of the time-delay system excited by Gaussian noise based on complex fractional moments.

In this paper, the transient response of the time-delay system under additive and multiplicative Gaussian white noise is investigated. Based on the approximate transformation method, we convert the time-delay system into an equivalent system without time delay. The one-dimensional Ito stochastic differential equation with respect to the amplitude response is derived by the stochastic averaging method, and Mellin transformation is utilized to transform the related Fokker-Planck-Kolmogorov equation in the real numbers field into a first-order ordinary differential equation (ODE) of complex fractional moments (CFM) in the complex number field. By solving the ODE of CFM, the transient probability density function can be constructed. Numerical methods are used to ascertain the effectiveness of the CFM method, the effects of system parameters on system response and the level of error vary with time as well as noise intensity are investigated. In addition, the CFM method is first implemented to analyze transient bifurcation, and the relation between CFM and bifurcation is discussed for the first time. Furthermore, the imperfect symmetry property appear on the projection map of joint probability density function.

A convolution based path integration method approach to the damped parametric pendulum under different random noise excitations

This paper is devoted to applying the path integration (PI) method to analyze the whole space dynamic behaviors of the damped parametric pendulum under different random noises. The random noises considered in this paper include external Gaussian white noise, external Poisson white noise and internal narrow-band noise, which are three typical approximated noises in Engineering. It’s a challenging and variable engineering task, since the system is non-homogenous, unbounded and may even lead to high dimensional cases. The convolution operation is an efficient way to accelerate the PI. However, the PI method with convolution operation is considered to have great limitation in handling the unbounded system under non-Gaussian noises. In order to extend the convolution operation to the unbounded rolling case, several special treatments in the back-stepping Runge-Kutta(RK) iterations, high space interpolations and the convolution operations are proposed. After above modification, a series of probability density functions(PDFs) of the system can be efficiently obtained via the convolution based PI method.Complexity analysis shows that the convolution based PI method is extremely efficient in obtaining the smooth transient PDF after reaching periodicity and 3-D stationary PDFs, while the Monte Carlo simulation(MCS) method is only suitable for calculating the time averaged PDF and high dimensional marginal PDF. With the help of the convolution based PI method, the influences of different noises on the damped parametric pendulum system are discussed comprehensively. The 3-D PDFs are novelly depicted in the form of point cloud map, which shows the convolution based PI a strong grasp of the details and the subtle information in the whole space PDF.

Dynamics of a coupled nonlinear energy harvester under colored noise and periodic excitations

Vibration energy harvester (VEH) has proven to be a favorable potential technique to supply continuous energy from ambient vibrations and its performance is greatly influenced by the design of potential structures. Motivated by the enhancement of energy harvesting performance, an electromechanical coupled VEH under colored noise and periodic excitations is investigated theoretically and numerically, and comparisons of the effects of mono-, bi- and tri-stable potentials are analyzed in detail. An uncoupled equivalent system, together with its joint probability density function (PDF), are derived by using the generalized harmonic transformation and the stochastic averaging based on energy-dependent frequency. For three kinds of different potentials, the effects of the periodic excitation, colored noise, potential shape and other crucial system parameters are explored on the dynamical behaviors. Results show that tri-stable VEH outperforms the mono- and bi-stable VEHs in mean harvested power and can achieve better stochastic resonance (SR) effect and higher power conversion efficiency by choosing the optimal system parameters, such as noise intensity, electromechanical coupling coefficient and time constant ratio. Moreover, the periodic excitation can change the topological characteristic of transient PDF, and larger periodic force can enhance SR effect but will weaken the power conversion efficiency. The theoretical results obtained by the proposed stochastic averaging method are well validated by numerical simulations.

First-passage-time distribution in a moving parabolic potential with spatial roughness.

In this paper, we investigate the first-passage-time distribution (FPTD) within a time-dependent parabolic potential in the presence of roughness with two methods: the Kramers theory and a nonsingular integral equation. By spatially averaging, the rough potential is equivalent to the combination of an effective smooth potential and an effective diffusion coefficient. Based on the Kramers theory, we first obtain Kramers approximations (KAs) of FPTD for both smooth and rough potentials. As expected, KA is valid only for high barriers and small external forces, and generally applicable for high barriers in rough potentials. To overcome the shortcoming of KA, a probability asymptotic approximation (PAA) based on an integral equation is proposed, which uses the transient probability density function (PDF) of the natural boundary conditions instead of the absorbing boundary conditions. We find that PAA fits very well even for large external forces. This enables us to analytically solve the FPTD for large external forces and low barriers as a strong extension to KA. In addition, we show that in the presence of a rough potential, the PAA of FPTD is in good agreement with numerical simulations for low barrier potentials. The PAA makes it possible to investigate the first-passage problem with ultrafast varying potentials and short exiting time. Thus, KA and PAA are complementary in determining the FPTD both for various barriers and external forces. Finally, the mean first-passage time (MFPT) is studied, which illustrates that the PAA of MFPT is effective almost in the whole range of external forces, while the KA of MFPT is valid only for small external forces.

Stochastic response and bifurcations of a dry friction oscillator with periodic excitation based on a modified short-time Gaussian approximation scheme

In this paper, a modified short-time Gaussian approximation (STGA) scheme is incorporated into the generalized cell mapping (GCM) method to study the stochastic response and bifurcations of a nonlinear dry friction oscillator with both periodic and Gaussian white noise excitations. Firstly, the Fokker–Planck–Kolmogorov equation and the moment equations with Gaussian closure are derived. Because of the periodic force and the singularity of moment equations in the initial condition being a Dirac delta function, a set of novel STGA solutions of the conditional probability density functions (PDFs) over each small fraction of one period is then constructed in order to form the solution over the whole period. At last, the transient and steady-state response PDFs are computed by the GCM method. The accuracy of the results is demonstrated by the direct Monte Carlo simulations. The transient PDFs are presented when evolving from a Gaussian initial distribution to a non-Gaussian steady-state one. The effects of periodic excitation and dry friction damping on the steady-state stochastic response are particularly discussed. When the amplitude of periodic excitation is changed, both stochastic P-bifurcation and D-bifurcation are detected. It is also found that the increase in the dry friction damping coefficient makes the chaotic response disappear gradually, inducing stochastic D-bifurcation but no stochastic P-bifurcation.

Local averaged path integration method approach for nonlinear dynamic systems

Traditional path integration method always has a limited time step length, whose minimum is based on the subdivision degree of integral space. In order to numerically analyze a smaller time step probability evolution without significant increase in computing time, a new local averaged path integration (LAPI) method is investigated. The basic idea of this method is giving a local averaged value of transition probability density function (TPDF). Then the numerical integration of Chapman–Kolmogorov (CK) equation to the local averaged probability density function (PDF) will be calculated in lower space subdivision degree. Second-order partial derivative of PDF can be used to correct the deviation between local averaged PDF and the real one. It is demonstrated that the proposed method can give a more efficient calculation compared with traditional Gauss–Legendre path integration (GLPI). In the case of our method, smaller time step value is permitted with the same space subdivision degree, and smaller space subdivision degree is needed to keep the accuracy of stationary PDF with the same time step value. This merits also exist when analyzing transient PDF, which leads to a better capability of capturing the evolution of the PDF.

Efficient statistically accurate algorithms for the Fokker–Planck equation in large dimensions

Solving the Fokker–Planck equation for high-dimensional complex turbulent dynamical systems is an important and practical issue. However, most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions (PDFs) of complex systems in turbulence, neuroscience and excitable media. In this article, efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of Fokker–Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures. The algorithms involve a hybrid strategy that requires only a small number of ensembles. Here, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious non-parametric Gaussian kernel density estimation in the remaining low-dimensional subspace. Particularly, the parametric method provides closed analytical formulae for determining the conditional Gaussian distributions in the high-dimensional subspace and is therefore computationally efficient and accurate. The full non-Gaussian PDF of the system is then given by a Gaussian mixture. Different from traditional particle methods, each conditional Gaussian distribution here covers a significant portion of the high-dimensional PDF. Therefore a small number of ensembles is sufficient to recover the full PDF, which overcomes the curse of dimensionality. Notably, the mixture distribution has significant skill in capturing the transient behavior with fat tails of the high-dimensional non-Gaussian PDFs, and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems. It is shown in a stringent set of test problems that the method only requires an order of O(100) ensembles to successfully recover the highly non-Gaussian transient PDFs in up to 6 dimensions with only small errors.

Time-dependent probability density function in cubic stochastic processes.

We report time-dependent probability density functions (PDFs) for a nonlinear stochastic process with a cubic force using analytical and computational studies. Analytically, a transition probability is formulated by using a path integral and is computed by the saddle-point solution (instanton method) and a new nonlinear transformation of time. The predicted PDF p(x,t) in general involves a time integral, and useful PDFs with explicit dependence on x and t are presented in certain limits (e.g., in the short and long time limits). Numerical simulations of the Fokker-Planck equation provide exact time evolution of the PDFs and confirm analytical predictions in the limit of weak noise. In particular, we show that transient PDFs behave drastically differently from the stationary PDFs in regard to the asymmetry (skewness) and kurtosis. Specifically, while stationary PDFs are symmetric with the kurtosis smaller than 3, transient PDFs are skewed with the kurtosis larger than 3; transient PDFs are much broader than stationary PDFs. We elucidate the effect of nonlinear interaction on the strong fluctuations and intermittency in the relaxation process.

Stochastic Response of Dynamical Systems with Fractional Derivative Term under Wide-Band Excitation

Transient solution of a fractional stochastic dynamical system under wide-band noise excitation is investigated. Generalized Harmonic Balance technique is firstly used to approximate restoring force of the given system as an amplitude-dependent form. In this way, stochastic averaging method then can be applied to transform the system into an Ito differential equation. Furthermore, the fractional derivative in the integral-differential form can be equivalent to a combination of periodic functions after the averaging procedure. As the following, Galerkin method therein is utilized to obtain the transient probability density functions by solving associated Fokker-Planck-Kolmogorov (FPK) equation. As an example, the Rayleigh oscillator is studied to illustrate the efficiency and accuracy of the proposed approaches. Numerical results show that exact stationary solution and transient solution derived from Galerkin method are in good agreement with those from Monte Carlo Simulation.

Stochastic response analysis of noisy system with non-negative real-power restoring force by generalized cell mapping method

The stochastic response of a noisy system with non-negative real-power restoring force is investigated. The generalized cell mapping (GCM) method is used to compute the transient and stationary probability density functions (PDFs). Combined with the global properties of the noise-free system, the evolutionary process of the transient PDFs is revealed. The results show that stochastic P-bifurcation occurs when the system parameter varies in the response analysis and the stationary PDF evolves from bimodal to unimodal along the unstable manifold during the bifurcation.

Nonstationary Response of Optimal Controlled Stochastic Van Der Pol Oscillator

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.

Modelling the dynamics of EU economic sentiment indicators: an interaction-based approach

This article estimates a simple univariate model of expectation or opinion formation in continuous time adapting a ‘canonical’ stochastic model of collective opinion dynamics (Weidlich and Haag, 1983; Lux, 1995, 2009a). This framework is applied to a selected data set on survey-based expectations from the rich EU business and consumer survey database for 12 European countries. The model parameters are estimated through Maximum Likelihood (ML) and numerical solution of the transient probability density functions for the resulting stochastic process. The model's success is assessed with respect to its out-of-sample forecasting performance relative to univariate Time Series (TS) models of the Autoregressive Moving Average model, ARMA(p, q) and Autoregressive Fractionally Integrated Moving Average, ARFIMA(p, d, q) varieties. These tests speak for a slight superiority of the canonical opinion dynamics model over the alternatives in the majority of cases.