Local galerkin method for the approximate solutions to general FPK equations

The Fokker–Planck–Kolmogorov (FPK) equation, as a well investigated partial differential equation, is of great significance to stochastic dynamics due to its theoretical rigor and exactness. However, practical difficulties with the FPK method are encountered when analysis of multi-degree-of-freedom (MDOF) systems with arbitrary nonlinearity is required. In the present paper, a cell renormalized method (CRM) which is based on a numerical determination of response statistical moments of the FPK equation is developed. Specifically, by invoking the concept of equivalence of probability flux, a cell renormalization procedure and a reconstruction scheme of derivative moments are introduced to divide the continuous state space into a discretized region of cells so that numerical derivative moments is derived. Subsequently, the Cell Renormalized FPK (CR-FPK) equation can be solved by a finite difference algorithm. Two numerical examples are included, and the effectiveness of the proposed method is assessed.

In this work, a novel shape control approach of the probability density function (PDF) for nonlinear stochastic systems is presented. First, we provide the formula for the PDF shape controller without devising the control law of the controller. Then, based on the exact analytical solution of the Fokker-Planck-Kolmogorov (FPK) equation, the product function of the polynomial and the exponential polynomial is regarded as the stationary PDF of the state response. To validate the performance of the proposed control approach, we compared it with the exponential polynomial method and the multi-Gaussian closure method by implementing comparative simulation experiments. The results show that the novel PDF shape control approach is effective and feasible. Using an equal number of parameters, our method can achieve a similar or better control effect as the exponential polynomial method. By comparison with the multi-Gaussian closure method, our method has clear advantages in PDF shape control performance. For all cases, the integral of squared error and the errors of first four moments of our proposed method were very small, indicating superior performance and promising good overall control effects of our method. The approach presented in this study provides an alternative for PDF shape control in nonlinear stochastic systems.

An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators

In this paper, the probabilistic solutions of the multi-degree-of-freedom (MDOF) or large-scale stochastic dynamic systems with polynomial type of nonlinearity and excited by Gaussian white noise excitations are obtained and investigated with the subspace method proposed recently by the authors. The space of the state variables of large-scale nonlinear stochastic dynamic (NSD) system excited by white noises is separated into two subspaces. Both sides of the Fokker–Planck–Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure (EPC) method. A simple flexural beam on nonlinear elastic springs is analyzed with the subspace method to show the effectiveness of the subspace-EPC method in this case.

In the present work, we consider an optimal control for a three-factor stochastic factor model. We assume that one of the factors is not observed and use classical filtering technique to transform the partial observation control problem for stochastic differential equation (SDE) to a full observation control problem for stochastic partial differential equation (SPDE). We then give a sufficient maximum principle for a system of controlled SDEs and degenerate SPDE. We also derive an equivalent stochastic maximum principle. We apply the obtained results to study a pricing and hedging problem of a commodity derivative at a given location, when the convenience yield is not observable.

An efficient method for solving high-dimension stationary FPK equation of strongly nonlinear systems under additive and/or multiplicative white noise

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

In this paper, the probability density function (PDF) and the mean up-crossing rate of nonlinear ship rolling in random beam seas are investigated. The excitation of stationary random sea waves is approximated as a second-order linear filtered white noise. The Fokker–Planck–Kolmogorov (FPK) equation governing the probability density function of ship rolling is a four-dimensional linear partial differential equation with varying coefficients, and obtaining its exact solution is much more sophisticated. The exponential-polynomial closure (EPC) method is applied to solve the corresponding FPK equation of the system. In numerical examples, linear-plus-cubic damping model and linear-plus-quadratic damping model with three different sea states are further examined. Comparison with the equivalent linearisation (EQL) method and Monte Carlo simulated results show that the proposed procedure is effective to obtain a satisfactory probability density function solution, especially in the tail region.

Random vibration of pretensioned rectangular membrane structures under heavy rainfall excitation

Infinite horizon optimal control problem of mean-field backward stochastic delay differential equation under partial information

The generation of continuous random processes with jointly specified probability density and covariation functions is considered. The proposed approach is based on the interpretation of the simulated process as a stationary output of a nonlinear dynamic system, excited by white Gaussian noise and described by a system of a first-order stochastic differential equations (SDE). The authors explore how the statistical characteristics of the equation's solution depends on the form of its operator and on the intensity of the input noise. Some aspects of the approximate synthesis of stochastic differential equations and examples of their application to the generation of non-Gaussian continuous processes are considered. The approach should be useful in signal processing when it is necessary to translate the available a priori information on the real random process into the language of its Markov model as well as in simulation of continuous correlated processes with the known probability density function. >

The parameter perturbation makes it difficult to analyze the dynamic mechanism of the brake chatter with the help of an ideal deterministic model. Hence, a stochastic dynamic model of brake chatter with the uncertainty of the torsional stiffness of the brake disc is established, the Stribeck model is used to describe the friction characteristic between the brake pad and brake disc. Then, the Itô stochastic differential equation is solved by means of the stochastic averaging method, the boundary type of the one-dimensional energy process is identified to discuss the stochastic stability of the brake system. Furthermore, the Fokker Planck Kolmogorov equation is derived, the dynamic bifurcation and phenomenological bifurcation are analytically proven, the influences of parameters, such as noise intensity, brake pressure, and friction coefficient, on bifurcation characteristics are revealed. The results show that the noise intensity, brake pressure, and friction coefficient difference need to be reduced to improve the system stability. Finally, the probability density function of the brake system is used to validate the analytical analysis. The relevant results provide a theoretical basis for better suppressing the brake chatter.

Probabilistic solutions of nonlinear oscillators excited by combined colored and white noise excitations

In the design of gas turbine engines, the analysis of nonlinear vibrations of mistuned and frictionally damped blade-disk assembly subjected to random excitation is highly complex. The transitional probability density function (PDF) for the random response of nonlinear systems under white or coloured noise excitation (delta-correlated) is governed by both the forward Fokker-Planck (FP) and backward Kolmogorov equations. This paper presents important improvement and extensions to a computationally efficient higher order, finite difference (FD) technique for the solution of higher dimensional FP equation corresponding to a two degree of freedom nonlinear system representative of vibration of tip shrouded frictionally damped bladed disk assembly subjected to Gaussian white noise excitation. Effects of friction damping on the mean square response of a blade are investigated. The friction coefficient of the damper is assumed to be a function of the sliding velocity of the contact surface. The effects of stiffness and damping mistuning on the forced response of frictionally damped bladed disk are investigated. Numerical studies are presented for a pair of mistuned blades of cyclic assemblies. The response and reliability of a blade subjected to random excitation is also obtained. With time averaged probability density as an invariant measure, the probability of large excursion in case of damping mistuning is also presented. The results of the FD method are validated by comparing with Monte Carlo Simulation (MCS) results.

We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time tau and provide a novel asymptotic estimate of the cumulative distribution function of tau . As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.