First-order Stochastic Differential Equations Research Articles

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.

Response of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitations: A Path Integral approach based on Laplace’s method of integration

In this paper, an approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional element and subjected to evolutionary excitations. This is achieved by a novel formulation of the Path Integral (PI) approach. Specifically, a stochastic averaging/linearization treatment of the original fractional order governing equation of motion yields a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is the Chapman–Kolmogorov (CK) equation governing the evolution in time of the non-stationary response amplitude PDF. Next, the PI technique is employed, which is based on a discretized version of the CK equation solved in short time steps. This is done relying on the Laplace’s method of integration which yields an approximate analytical solution of the integral involved in the CK equation. In this manner, the repetitive integrations generally required in the classical numerical implementation of the procedure are avoided. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. Notably, the technique can also account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. Applications to oscillators with Van der Pol and Duffing type nonlinear restoring force models, and Preisach hysteretic models, are presented. Appropriate comparisons with Monte Carlo simulation data are shown, demonstrating the efficiency and accuracy of the proposed approach.

Stochastic Volterra integral equations and a class of first-order stochastic partial differential equations

We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued Lévy noise and integration kernels may have non-linear dependence on the current state of the process. Our method is based on an embedding into a Hilbert space of functions which allows to represent the solution of the Volterra equation as the boundary value of a solution to a stochastic partial differential equation. We first gather abstract results and give more detailed conditions in more specific function spaces.

First-passage time study of a stochastic growth model

In this paper, a stochastic nonlinear growth model is proposed, which can be considered a generalization of the stochastic logistic model. It can be applied to the growth of a generic population as well as to the propagation of the fracture in engineering materials. The excitation is assumed to be a stationary Gaussian white noise stochastic process, which affects the system parametrically. A preliminary study of the stochastic differential equation governing the model reveals that there is a phase transition when the nonlinearity parameter c reaches one: when $$c<1$$ , the system tends to the stationary state, while it is never stationary when $$c\ge 1$$ . Then, attention is focused on the first-passage time problem, which is of crucial importance for dynamical systems. The first-order stochastic differential equation (SDE) that describes the model is transformed into an Ito’s SDE by adding the Wong–Zakai–Stratonovich corrective term. For the last equation, the backward Kolmogorov equation is formulated. By solving it with appropriate initial and boundary conditions, the probability of survival is obtained, that is, the probability of not exceeding a given threshold. The solution is looked for three cases $$c 1$$ . In any case, the numerical analyses show that the survival probability decays fast.

An approximate technique for determining in closed form the response transition probability density function of diverse nonlinear/hysteretic oscillators

An approximate analytical technique is developed for determining, in closed form, the transition probability density function (PDF) of a general class of first-order stochastic differential equations (SDEs) with nonlinearities both in the drift and in the diffusion coefficients. Specifically, first, resorting to the Wiener path integral most probable path approximation and utilizing the Cauchy–Schwarz inequality yields a closed-form expression for the system response PDF, at practically zero computational cost. Next, the accuracy of this approximation is enhanced by proposing a more general PDF form with additional parameters to be determined. This is done by relying on the associated Fokker–Planck operator to formulate and solve an error minimization problem. Besides the mathematical merit of the derived closed-form approximate PDFs, an additional significant advantage of the technique relates to the fact that it can be readily coupled with a stochastic averaging treatment of second-order SDEs governing the dynamics of diverse stochastically excited nonlinear/hysteretic oscillators. In this regard, it is shown that the technique is capable of determining approximately the response amplitude transition PDF of a wide range of nonlinear oscillators, including hysteretic systems following the Preisach versatile modeling. Several numerical examples are considered for demonstrating the reliability and computational efficiency of the technique. Comparisons with pertinent Monte Carlo simulation data are provided as well.

Non-stationary response statistics of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional derivative elements and subjected to evolutionary stochastic excitation. Specifically, resorting to stochastic averaging/linearization leads to a dimension reduction of the governing equation of motion and to a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is a Fokker–Planck partial differential equation governing the evolution in time of the non-stationary response amplitude PDF. Next, assuming an appropriately chosen time-dependent PDF form of the Rayleigh kind for the response amplitude, and substituting into the Fokker–Planck equation, yields a deterministic first-order nonlinear ordinary differential equation for the time-dependent PDF coefficient. This can be readily solved numerically via standard deterministic integration schemes. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. The technique can account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. A hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative elements are considered in the numerical examples section. To assess the accuracy of the developed technique, the analytical results are compared with pertinent Monte Carlo simulation data.

Flicker Noise Simulation by Superposition of Normal Stationary Processes

The details of the well-known method for representing 1/f noise by a sum of stationary processes are presented. Two series of shaping filters are applied in the frequency band specified by the user. The relations for the filter parameters are determined. The error in the approximation of the spectrum is considered. An algorithm for modeling fluctuations in the form of a system of first-order stochastic difference equations is proposed. The algorithm is intended to be used in simulation models applied to study the errors of navigation systems.

Fokker-Planck equation of the reduced Wigner function associated to an Ohmic quantum Langevin dynamics.

This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.

Dynamic Reconstruction of Disturbances in a Quasilinear Stochastic Differential Equation

The problem of reconstructing unknown inputs in a first-order quasilinear stochastic differential equation is studied by applying dynamic inversion theory. The disturbances in the deterministic and stochastic terms of the equation are simultaneously reconstructed using discrete information on some realizations of the stochastic process. The problem is reduced to an inverse one for ordinary differential equations satisfied by the expectation and variance of the original process. A finite-step software implementable solution algorithm is proposed, and its accuracy with respect to the number of measured realizations is estimated. An illustrative example is given.

Stationary Solution of Duffing Oscillator Driven by Additive and Multiplicative Colored Noise Excitations

A bistable Duffing oscillator subjected to additive and multiplicative Ornstein–Uhlenbeck (OU) colored excitations is examined. It is modeled through a set of four first-order stochastic differential equations by representing the OU excitations as filtered Gaussian white noise excitations. Enlargement in the state-space vector leads to four-dimensional (4D) Fokker–Planck–Kolmogorov (FPK) equation. The exponential-polynomial closure (EPC) method, proposed previously for the case of white noise excitations, is further improved and developed to solve colored noise case, resulting in much more polynomial terms included in the approximate solution. Numerical results show that approximate solutions from the EPC method compare well with the predictions obtained via Monte Carlo simulation (MCS) method. Investigation is also carried out to examine the influence of intensity level on the probability distribution solutions of system responses.

Unification and simplification for position updating formulas in particle swarm optimization

Particle swarm optimization (PSO) has been a popular research area in artificial intelligence technology, where the two issues of theoretical analysis and premature convergence have been the focus of attention. However, due to the complex dynamics of a particle swarm, the former was conducted only for simplified systems. The latter has only been addressed by introducing some additional operations, which inevitably increases the complexity of PSO and complicates the theoretical analysis. This paper proposes a unified and simplified rule for position updating in the existing algorithms as an attempt to solve the above-mentioned problems. This rule simplifies the multiple-order stochastic difference equation to a first-order stochastic difference equation, and facilitates the analysis of the convergence and control of the search behavior of particles. Experiments were conducted on some representative algorithms, and the results verified the correctness of the unification and simplification of position updating formulas, which also performed more competitively.

Simple Generation of Gamma, Gamma–Gamma, and K Distributions With Exponential Autocorrelation Function

The simulation of a free-space optical (FSO) communication channel in the presence of strong turbulence typically requires the generation of channel states with a K or gamma–gamma distribution and a predefined autocorrelation function. In this paper, we propose a simple and effective simulator of the strong-turbulence FSO channel that addresses the influence of the temporal covariance effect. Specifically, the proposed simulator provides K and gamma–gamma distributed values with the exponential autocorrelation function and a prescribed correlation time. This simulator is based on the numerical solution of the first-order stochastic differential equation. The simulated channel states are generated by a simple discrete-time differential equation and the simulator performance is analyzed in the paper.

Free-space optical channel simulator for weak-turbulence conditions.

Free-space optical (FSO) communication may be severely influenced by the inevitable turbulence effect that results in channel gain fluctuations and fading. The objective of this paper is to provide a simple and effective simulator of the weak-turbulence FSO channel that emulates the influence of the temporal covariance effect. Specifically, the proposed model is based on lognormal distributed samples with a corresponding correlation time. The simulator is based on the solution of the first-order stochastic differential equation (SDE). The results of the provided SDE analysis reveal its efficacy for turbulent channel modeling.

Newton’s method for first-order stochastic functional partial differential equations

We apply Newton’s method to hyperbolic stochastic functional partial differential equations of the first order driven by a multidimensional Brownian motion. We prove a first-order convergence and a second-order convergence in a probabilistic sense.

A Rain-Attenuation Stochastic Dynamic Model for LEO Satellite Systems Above 10 GHz

In this paper, a new rain-attenuation stochastic dynamic channel model for low Earth orbit (LEO) satellite links operating above 10 GHz is proposed, taking into account the time dependence of elevation angle and its impact on rain-attenuation dynamics. The new synthesizer is based on the first-order stochastic differential equations (SDEs) and extends the well-known Maseng–Bakken (M–B) model for fixed satellite communication links. Moreover, an analytical closed-form expression of the rain-attenuation dynamic parameter is derived and used in the proposed synthesizer. The new channel model is validated in terms of the LEO-slant-path rain-attenuation exceedance probability. Finally, a new analytical model for the calculation of fade-slope conditional exceedance probability is presented, which is then validated with accurate simulations. The new models are directly applicable for the evaluation and design of fade mitigation techniques (FMTs) in LEO satellite systems operating at frequencies above 10 GHz.

Finite element solution of Fokker–Planck equation of nonlinear oscillators subjected to colored non-Gaussian noise

Nonlinear oscillators subjected to colored Gaussian/non-Gaussian excitations are modelled through a set of three coupled first-order stochastic differential equations by representing the excitation as a first-order filtered white noise. A 3-D finite element (FE) formulation is developed to solve the corresponding 3-D Fokker Planck (FP) equations. The joint probability density functions of the state variables, obtained as a solution of the FP equation, are typically non-Gaussian and are used for computing the crossing statistics of the response – an essential metric for time variant reliability analysis. The method is illustrated through a noisy Lorenz attractor and a Duffing oscillator subjected to additive colored noise. The increase in state-space dimension when the Duffing oscillator is additionally excited with a parametric Gaussian noise is effectively handled by using stochastic averaging to reduce the state-space dimension. Investigations are carried out to examine the accuracy of the FE method vis-a-vis Monte Carlo simulations. The proposed method is observed to be computationally significantly cheaper for these three problems.

Channel Model for Satellite Communication Links Above 10GHz Based on Weibull Distribution

Modern satellite communication networks will employ frequencies above 10GHz. At these frequency bands, rain attenuation is the dominant fading mechanism. In this paper, a novel channel model, a synthesizer for generating rain attenuation time series for satellite links operating at 10GHz and above is presented. The proposed channel model modifies Maseng-Bakken (M-B) model since it generates rain attenuation time series that follow the Weibull distribution. The new stochastic dynamic model is based on the first-order Stochastic Differential Equations (SDEs) and considers rain attenuation induced on a slant path as a Weibull-based stochastic process. Moreover, the theoretical expressions for the computation of the exceedance probability of hitting time random variable are presented. The hitting time statistics may be employed for the optimum design of Fade Mitigation Techniques (FMTs). The synthesizer is verified in terms of the exceedance probability and the theoretical CCDF of hitting time comparing to these derived from the simulations in the numerical results section.

On the Exact Discretization of a Continuous Time AR(1) Model driven by either Long Memory or Antipersistent Innovations: A Fractional Algebra Approach

AbstractExact discretization formulae are established for a first-order stochastic differential equation driven by a fractional noise of either long memory or antipersistent type. We assume that the underlying process is sampled at non-unit equispaced observational intervals. Using fractional algebra techniques the exact discretization formulae are derived in terms of confluent hypergeometric and incomplete gamma functions which admit infinite order series expansions.

Response and First-Passage Statistics of Nonlinear Oscillators via a Numerical Path Integral Approach

AbstractA numerical path integral solution approach is developed for determining the response and first-passage probability density functions (PDFs) of nonlinear oscillators subject to evolutionary broad-band stochastic excitations. Specifically, based on the concepts of statistical linearization and of stochastic averaging, the system response amplitude is modeled as a one-dimensional Markov diffusion process. Further, using a discrete version of the Chapman-Kolmogorov equation and the associated first-order stochastic differential equation, the response amplitude and first-passage PDFs are derived. The main concept of the approach relates to the evolution of the response PDF in short time steps, assuming a Gaussian form for the conditional response PDF. A number of nonlinear oscillators are considered in the numerical examples section including the versatile Preisach hysteretic oscillator. For this oscillator, first-passage PDFs are derived for the first time to the authors’ best knowledge. Comparisons ...

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

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of a class of randomly excited nonlinear oscillators is developed. Specifically, combining the concepts of statistical linearization and of stochastic averaging the evolution of the response amplitude of oscillators with nonlinear damping is captured by a first-order stochastic differential equation (SDE). This equation has nonlinear drift but constant diffusion coefficients. This convenient feature of the SDE along with the concept of the Wiener path integral is utilized in conjunction with a variational formulation to derive an approximate closed form solution for the response amplitude PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes. In this manner, an analytical Wiener path integral based technique is developed for treating certain stochastic dynamics problems for the first time. Further, the technique can be used as a convenient tool for assessing the accuracy of alternative, more general, approximate solution methods. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.