The paper studies the parametric stochastic roll motion in the random waves.The differential equation of the ship parametric roll under random wave is established with considering the nonlinear damping and ship speed....The paper studies the parametric stochastic roll motion in the random waves.The differential equation of the ship parametric roll under random wave is established with considering the nonlinear damping and ship speed.Random sea surface is treated as a narrow-band stochastic process,and the stochastic parametric excitation is studied based on the effective wave theory.The nonlinear restored arm function obtained from the numerical simulation is expressed as the approximate analytic function.By using the stochastic averaging method,the differential equation of motion is transformed into Ito’s stochastic differential equation.The steady-state probability density function of roll motion is obtained,and the results are validated with the numerical simulation and model test.展开更多
In this article,we investigate a stochastic Galerkin method for the Maxwell equations with random inputs.The generalized Polynomial Chaos(gPC)expansion technique is used to obtain a deterministic system of the gPC exp...In this article,we investigate a stochastic Galerkin method for the Maxwell equations with random inputs.The generalized Polynomial Chaos(gPC)expansion technique is used to obtain a deterministic system of the gPC expansion coefficients.The regularity of the solution with respect to the random is analyzed.On the basis of the regularity results,the optimal convergence rate of the stochastic Galerkin approach for Maxwell equations with random inputs is proved.Numerical examples are presented to support the theoretical analysis.展开更多
In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochas...In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochastic differential equations (SDE), therefore the Fokker Planck Kolmogorov (FPK) equation is expressed in general form with no limitation on the degree of nonlinearity of the SDE, the type of δ correlated excitations, the existence of multiplicative excitations, and the dimension of SDE or FPK equation. Examples are given and numerical results are provided for comparing with known exact solution to show the effectiveness of the method.展开更多
In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in R^d (d = 2, 3). Convergence analysis and error estimates are presented for the numerica...In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in R^d (d = 2, 3). Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.展开更多
We develop a fast stochastic Galerkin method for an optimal control problem governed by a random space-fractional diffusion equation with deterministic constrained control. Optimal control problems governed by a fract...We develop a fast stochastic Galerkin method for an optimal control problem governed by a random space-fractional diffusion equation with deterministic constrained control. Optimal control problems governed by a fractional diffusion equation tends to provide a better description for transport or conduction processes in heterogeneous media. Howev- er, the fractional control problem introduces significant computation complexity due to the nonlocal nature of fractional differential operators, and this is further worsen by the large number of random space dimensions to discretize the probability space. We ap- proximate the optimality system by a gradient algorithm combined with the stochastic Galerkin method through the discretization with respect to both the spatial space and the probability space. The resulting linear system can be decoupled for the random and spatial variable, and thus solved separately. A fast preconditioned Bi-Conjugate Gradient Stabilized method is developed to efficiently solve the decoupled systems derived from the fractional diffusion operators in the spatial space. Numerical experiments show the utility of the method.展开更多
In an interdisciplinary field on mathematics and physics,we examine a physical problem,fluid flow in porous media,which is represented by a stochastic partial differential equation(SPDE).We first give a priori error e...In an interdisciplinary field on mathematics and physics,we examine a physical problem,fluid flow in porous media,which is represented by a stochastic partial differential equation(SPDE).We first give a priori error estimates for the solutions to an optimization problem constrained by the physical model under lower regularity assumptions than the literature.We then use the concept of Galerkin finite element methods to establish a new numerical algorithm to give approximations for our stochastic optimal physical problem.Finally,we develop original computer programs based on the algorithm and use several numerical examples of various situations to see how well our solver works by comparing its outputs to the priori error estimates.展开更多
In this paper, we investigate an SIS model with treatment and immigration. Firstly, the two-dimensional model is simplified by using the stochastic averaging method. Then, we derive the local stability of the stochast...In this paper, we investigate an SIS model with treatment and immigration. Firstly, the two-dimensional model is simplified by using the stochastic averaging method. Then, we derive the local stability of the stochastic system by computing the Lyapunov exponent of the linearized system. Further, the global stability of the stochastic model is analyzed based on the singular boundary theory. Moreover, we prove that the model undergoes a Hopf bifurcation and a pitchfork bifurcation. Finally, several numerical examples are provided to illustrate the theoretical results. .展开更多
Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response sta-tistics.In the present pa...Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response sta-tistics.In the present paper,the basic idea and procedure of the stochastic averaging method for quasi Hamiltonian systems are briefly introduced.The applications of the stochastic averaging method in studying the dynamics of active Brownian particles,the reaction rate theory,the dynamics of breathing and denaturation of DNA,and the Fermi resonance and its effect on the mean transition time are reviewed.展开更多
This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay dif...This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay differential equations.We prove that the discontinuous Galerkin scheme is strongly convergent:globally stable and analogously asymptotically stable in mean square sense.In addition,this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential equations.Numerical tests indicate that the method has first-order and half-order strong mean square convergence,when the diffusion term is without delay and with delay,respectively.展开更多
This study presents a new tool for solving stochastic boundary-value problems. This tool is created by modify the previous spectral stochastic meshless local Petrov-Galerkin method using the MLPG5 scheme. This modifie...This study presents a new tool for solving stochastic boundary-value problems. This tool is created by modify the previous spectral stochastic meshless local Petrov-Galerkin method using the MLPG5 scheme. This modified spectral stochastic meshless local Petrov-Galerkin method is selectively applied to predict the structural failure probability with the uncertainty in the spatial variability of mechanical properties. Except for the MLPG5 scheme, deriving the proposed spectral stochastic meshless local Petrov-Galerkin formulation adopts generalized polynomial chaos expansions of random mechanical properties. Predicting the structural failure probability is based on the first-order reliability method. Further comparing the spectral stochastic finite element-based and meshless local Petrov-Galerkin-based predicted structural failure probabilities indicates that the proposed spectral stochastic meshless local Petrov-Galerkin method predicts the more accurate structural failure probability than the spectral stochastic finite element method does. In addition, generating spectral stochastic meshless local Petrov-Galerkin results are considerably time-saving than generating Monte-Carlo simulation results does. In conclusion, the spectral stochastic meshless local Petrov-Galerkin method serves as a time-saving tool for solving stochastic boundary-value problems sufficiently accurately.展开更多
A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise (fGn) with the Hurst index 1/2 〈 H 〈 1 is proposed. First, the definition and the ba...A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise (fGn) with the Hurst index 1/2 〈 H 〈 1 is proposed. First, the definition and the basic property of fGn and related fractional Brownian motion (iBm) are briefly introduced. Then, the averaged fractional stochastic differential equations (SDEs) for the first integrals and combinations of angle variables of the associated Hamiltonian systems are derived. The stationary probability density and statistics of the original systems are then obtained approximately by simulating the averaged SDEs numerically. An example is worked out to illustrate the proposed stochastic averaging method. It is shown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of original system agree well.展开更多
This paper is concerned with preconditioners for interior penalty discontinuous Galerkin discretizations of second-order elliptic boundary value problems.We extend earlier related results in[7]in the following sense.S...This paper is concerned with preconditioners for interior penalty discontinuous Galerkin discretizations of second-order elliptic boundary value problems.We extend earlier related results in[7]in the following sense.Several concrete realizations of splitting the nonconforming trial spaces into a conforming and(remaining)nonconforming part are identified and shown to give rise to uniformly bounded condition numbers.These asymptotic results are complemented by numerical tests that shed some light on their respective quantitative behavior.展开更多
In power systems, there are many uncertainty factors such as power outputs of distributed generations and fluctuations of loads. It is very beneficial to power system analysis to acquire an explicit function describin...In power systems, there are many uncertainty factors such as power outputs of distributed generations and fluctuations of loads. It is very beneficial to power system analysis to acquire an explicit function describing the relationship between these factors(namely parameters) and power system states(or performances). This problem, termed as parametric problem(PP) in this paper, can be solved by Galerkin method,which is a powerful and mathematically rigorous method aiming to seek an accurate explicit approximate function by projection techniques. This paper provides a review of the applications of polynomial approximation based on Galerkin method in power system PPs as well as stochastic problems. First, the fundamentals of polynomial approximation and Galerkin method are introduced. Then, the process of solving three types of typical PPs by polynomial approximation based on Galerkin method is elaborated. Finally, some application examples as well as several potential applications of power system PPs solved by Galerkin method are presented, namely the probabilistic power flow, approximation of static voltage stability region boundary, and parametric time-domain dynamic simulation.展开更多
A stochastic averaging method for predicting the response of quasi partially integrable and non-resonant Hamiltoniansystems to fractional Gaussian noise (fGla) with the Hurst index 1/2〈H〈l is proposed. The average...A stochastic averaging method for predicting the response of quasi partially integrable and non-resonant Hamiltoniansystems to fractional Gaussian noise (fGla) with the Hurst index 1/2〈H〈l is proposed. The averaged stochastic differential equa-tions (SDEs) for the first integrals of the associated Hamiltonian system are derived. The dimension of averaged SDEs is less thanthat of the original system. The stationary probability density and statistics of the original system are obtained approximately fromsolving the averaged SDEs numerically. Two systems are worked out to illustrate the proposed stochastic averaging method. It isshown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of originalsystem agree well, and the computational time for the former results is less than that for the latter ones.展开更多
In this paper, we study the p-order cone constraint stochastic variational inequality problem. We first take the sample average approximation method to deal with the expectation and gain an approximation problem, furt...In this paper, we study the p-order cone constraint stochastic variational inequality problem. We first take the sample average approximation method to deal with the expectation and gain an approximation problem, further the rationality is given. When the underlying function is Lipschitz continuous, we acquire a projection and contraction algorithm to solve the approximation problem. In the end, the method is applied to some numerical experiments and the effectiveness of the algorithm is verified.展开更多
A stochastic wheelset model with a nonlinear wheel-rail contact relationship is established to investigate the stochastic stability and stochastic bifurcation of the wheelset system with the consideration of the stoch...A stochastic wheelset model with a nonlinear wheel-rail contact relationship is established to investigate the stochastic stability and stochastic bifurcation of the wheelset system with the consideration of the stochastic parametric excitations of equivalent conicity and suspension stiffness.The wheelset is systematized into a onedimensional(1D)diffusion process by using the stochastic average method,the behavior of the singular boundary is analyzed to determine the hunting stability condition of the wheelset system,and the critical speed of stochastic bifurcation is obtained.The stationary probability density and joint probability density are derived theoretically.Based on the topological structure change of the probability density function,the stochastic Hopf bifurcation form and bifurcation condition of the wheelset system are determined.The effects of stochastic factors on the hunting stability and bifurcation characteristics are analyzed,and the simulation results verify the correctness of the theoretical analysis.The results reveal that the boundary behavior of the diffusion process determines the hunting stability of the stochastic wheelset system,and the left boundary characteristic value cL=1 is the critical state of hunting stability.Besides,stochastic D-bifurcation and P-bifurcation will appear in the wheelset system,and the critical speeds of the two kinds of stochastic bifurcation decrease with the increase in the stochastic parametric excitation intensity.展开更多
This paper focuses on the stochastic analysis of a viscoelastic bistable energy harvesting system under colored noise and harmonic excitation, and adopts the time-delayed feedback control to improve its harvesting eff...This paper focuses on the stochastic analysis of a viscoelastic bistable energy harvesting system under colored noise and harmonic excitation, and adopts the time-delayed feedback control to improve its harvesting efficiency. Firstly, to obtain the dimensionless governing equation of the system, the original bistable system is approximated as a system without viscoelastic term by using the stochastic averaging method of energy envelope, and then is further decoupled to derive an equivalent system. The credibility of the proposed method is validated by contrasting the consistency between the numerical and the analytical results of the equivalent system under different noise conditions. The influence of system parameters on average output power is analyzed, and the control effect of the time-delayed feedback control on system performance is compared. The output performance of the system is improved with the occurrence of stochastic resonance(SR). Therefore, the signal-to-noise ratio expression for measuring SR is derived, and the dependence of its SR behavior on different parameters is explored.展开更多
The approximate transient response of quasi in- tegrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged It6 equa- tions for independent motion integrals and the associa...The approximate transient response of quasi in- tegrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged It6 equa- tions for independent motion integrals and the associated Fokker-Planck-Kolmogorov (FPK) equation governing the transient probability density of independent motion integrals of the system are derived by applying the stochastic averag- ing method for quasi integrable Hamiltonian systems. Then, approximate solution of the transient probability density of independent motion integrals is obtained by applying the Galerkin method to solve the FPK equation. The approxi- mate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coeffi- cients. The transient probability densities of displacements and velocities can be derived from that of independent mo- tion integrals. Three examples are given to illustrate the ap- plication of the proposed procedure. It is shown that the re- suits for the three examples obtained by using the proposed procedure agree well with those from Monte Carlo simula- tion of the original systems.展开更多
We studied the response of fractional-order van de Pol oscillator to Gaussian white noise excitation in this letter. An equivalent integral-order nonlinear stochastic system is obtained to replace the given system bas...We studied the response of fractional-order van de Pol oscillator to Gaussian white noise excitation in this letter. An equivalent integral-order nonlinear stochastic system is obtained to replace the given system based on the principle of minimum mean-square error. Through stochastic averaging, an averaged Ito equation is deduced. We obtained the Fokker–Planck–Kolmogorov equation connected to the averaged Ito equation and solved it to yield the approximate stationary response of the system. The analytical solution is confirmed by using Monte Carlo simulation.展开更多
基金the State Administration of Science,Technology and Industry for National Defense of China(Grant No.B2420132001).
文摘The paper studies the parametric stochastic roll motion in the random waves.The differential equation of the ship parametric roll under random wave is established with considering the nonlinear damping and ship speed.Random sea surface is treated as a narrow-band stochastic process,and the stochastic parametric excitation is studied based on the effective wave theory.The nonlinear restored arm function obtained from the numerical simulation is expressed as the approximate analytic function.By using the stochastic averaging method,the differential equation of motion is transformed into Ito’s stochastic differential equation.The steady-state probability density function of roll motion is obtained,and the results are validated with the numerical simulation and model test.
基金Supported by NSFC (91430107/11771138/11171104)the Construct Program of the Key Discipline in Hunan+4 种基金partially supported by Scientific Research Fund of Hunan Provincial Education Department (19B325/19C1059)Hunan International Economics University (2017A05)supported by NSFC (11771137)the Construct Program of the Key Discipline in Hunan Provincea Scientific Research Fund of Hunan Provincial Education Department (16B154)。
文摘In this article,we investigate a stochastic Galerkin method for the Maxwell equations with random inputs.The generalized Polynomial Chaos(gPC)expansion technique is used to obtain a deterministic system of the gPC expansion coefficients.The regularity of the solution with respect to the random is analyzed.On the basis of the regularity results,the optimal convergence rate of the stochastic Galerkin approach for Maxwell equations with random inputs is proved.Numerical examples are presented to support the theoretical analysis.
文摘In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochastic differential equations (SDE), therefore the Fokker Planck Kolmogorov (FPK) equation is expressed in general form with no limitation on the degree of nonlinearity of the SDE, the type of δ correlated excitations, the existence of multiplicative excitations, and the dimension of SDE or FPK equation. Examples are given and numerical results are provided for comparing with known exact solution to show the effectiveness of the method.
基金NSF under grant number 0609918AFOSR under grant numbers FA9550-06-1-0234 and FA9550-07-1-0154+2 种基金NSFC (10671082,10626026,10471054)NNSF (No.10701039 of China)985 program of Jilin University
文摘In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in R^d (d = 2, 3). Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.
基金This work was supported by the National Natural Science Foundation of China under grants 11371229, 11571026 and 11501326, and by the China Scholarship Council (File No. 2013083Y0102).
文摘We develop a fast stochastic Galerkin method for an optimal control problem governed by a random space-fractional diffusion equation with deterministic constrained control. Optimal control problems governed by a fractional diffusion equation tends to provide a better description for transport or conduction processes in heterogeneous media. Howev- er, the fractional control problem introduces significant computation complexity due to the nonlocal nature of fractional differential operators, and this is further worsen by the large number of random space dimensions to discretize the probability space. We ap- proximate the optimality system by a gradient algorithm combined with the stochastic Galerkin method through the discretization with respect to both the spatial space and the probability space. The resulting linear system can be decoupled for the random and spatial variable, and thus solved separately. A fast preconditioned Bi-Conjugate Gradient Stabilized method is developed to efficiently solve the decoupled systems derived from the fractional diffusion operators in the spatial space. Numerical experiments show the utility of the method.
基金supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)(grant number NRF 2010-0026032)supported by a faculty research grant from the University of Mary Washington.
文摘In an interdisciplinary field on mathematics and physics,we examine a physical problem,fluid flow in porous media,which is represented by a stochastic partial differential equation(SPDE).We first give a priori error estimates for the solutions to an optimization problem constrained by the physical model under lower regularity assumptions than the literature.We then use the concept of Galerkin finite element methods to establish a new numerical algorithm to give approximations for our stochastic optimal physical problem.Finally,we develop original computer programs based on the algorithm and use several numerical examples of various situations to see how well our solver works by comparing its outputs to the priori error estimates.
文摘In this paper, we investigate an SIS model with treatment and immigration. Firstly, the two-dimensional model is simplified by using the stochastic averaging method. Then, we derive the local stability of the stochastic system by computing the Lyapunov exponent of the linearized system. Further, the global stability of the stochastic model is analyzed based on the singular boundary theory. Moreover, we prove that the model undergoes a Hopf bifurcation and a pitchfork bifurcation. Finally, several numerical examples are provided to illustrate the theoretical results. .
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10772159 and 10802074)the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20060335125)the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y7080070)
文摘Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response sta-tistics.In the present paper,the basic idea and procedure of the stochastic averaging method for quasi Hamiltonian systems are briefly introduced.The applications of the stochastic averaging method in studying the dynamics of active Brownian particles,the reaction rate theory,the dynamics of breathing and denaturation of DNA,and the Fermi resonance and its effect on the mean transition time are reviewed.
基金the National Natural Science Foundation of China(No.11671343).
文摘This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay differential equations.We prove that the discontinuous Galerkin scheme is strongly convergent:globally stable and analogously asymptotically stable in mean square sense.In addition,this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential equations.Numerical tests indicate that the method has first-order and half-order strong mean square convergence,when the diffusion term is without delay and with delay,respectively.
文摘This study presents a new tool for solving stochastic boundary-value problems. This tool is created by modify the previous spectral stochastic meshless local Petrov-Galerkin method using the MLPG5 scheme. This modified spectral stochastic meshless local Petrov-Galerkin method is selectively applied to predict the structural failure probability with the uncertainty in the spatial variability of mechanical properties. Except for the MLPG5 scheme, deriving the proposed spectral stochastic meshless local Petrov-Galerkin formulation adopts generalized polynomial chaos expansions of random mechanical properties. Predicting the structural failure probability is based on the first-order reliability method. Further comparing the spectral stochastic finite element-based and meshless local Petrov-Galerkin-based predicted structural failure probabilities indicates that the proposed spectral stochastic meshless local Petrov-Galerkin method predicts the more accurate structural failure probability than the spectral stochastic finite element method does. In addition, generating spectral stochastic meshless local Petrov-Galerkin results are considerably time-saving than generating Monte-Carlo simulation results does. In conclusion, the spectral stochastic meshless local Petrov-Galerkin method serves as a time-saving tool for solving stochastic boundary-value problems sufficiently accurately.
基金supported by the National Natural Science Foundation of China under grants nos.:11272279,11321202 and 11432012
文摘A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise (fGn) with the Hurst index 1/2 〈 H 〈 1 is proposed. First, the definition and the basic property of fGn and related fractional Brownian motion (iBm) are briefly introduced. Then, the averaged fractional stochastic differential equations (SDEs) for the first integrals and combinations of angle variables of the associated Hamiltonian systems are derived. The stationary probability density and statistics of the original systems are then obtained approximately by simulating the averaged SDEs numerically. An example is worked out to illustrate the proposed stochastic averaging method. It is shown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of original system agree well.
基金This work has been supported in part by the French-German PROCOPE contract 11418YBby the European Commission Human Potential Programme under contract HPRN-CT-2002-00286“Breaking Complexity”,by the SFB 401 and the Leibniz Pro-gramme funded by DFG.
文摘This paper is concerned with preconditioners for interior penalty discontinuous Galerkin discretizations of second-order elliptic boundary value problems.We extend earlier related results in[7]in the following sense.Several concrete realizations of splitting the nonconforming trial spaces into a conforming and(remaining)nonconforming part are identified and shown to give rise to uniformly bounded condition numbers.These asymptotic results are complemented by numerical tests that shed some light on their respective quantitative behavior.
基金supported by the National Natural Science Foundation of China (No. 51777184)。
文摘In power systems, there are many uncertainty factors such as power outputs of distributed generations and fluctuations of loads. It is very beneficial to power system analysis to acquire an explicit function describing the relationship between these factors(namely parameters) and power system states(or performances). This problem, termed as parametric problem(PP) in this paper, can be solved by Galerkin method,which is a powerful and mathematically rigorous method aiming to seek an accurate explicit approximate function by projection techniques. This paper provides a review of the applications of polynomial approximation based on Galerkin method in power system PPs as well as stochastic problems. First, the fundamentals of polynomial approximation and Galerkin method are introduced. Then, the process of solving three types of typical PPs by polynomial approximation based on Galerkin method is elaborated. Finally, some application examples as well as several potential applications of power system PPs solved by Galerkin method are presented, namely the probabilistic power flow, approximation of static voltage stability region boundary, and parametric time-domain dynamic simulation.
基金supported by the National Natural Science Foundation of China(Nos.11172259,11272279,11321202,and 11432012)
文摘A stochastic averaging method for predicting the response of quasi partially integrable and non-resonant Hamiltoniansystems to fractional Gaussian noise (fGla) with the Hurst index 1/2〈H〈l is proposed. The averaged stochastic differential equa-tions (SDEs) for the first integrals of the associated Hamiltonian system are derived. The dimension of averaged SDEs is less thanthat of the original system. The stationary probability density and statistics of the original system are obtained approximately fromsolving the averaged SDEs numerically. Two systems are worked out to illustrate the proposed stochastic averaging method. It isshown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of originalsystem agree well, and the computational time for the former results is less than that for the latter ones.
文摘In this paper, we study the p-order cone constraint stochastic variational inequality problem. We first take the sample average approximation method to deal with the expectation and gain an approximation problem, further the rationality is given. When the underlying function is Lipschitz continuous, we acquire a projection and contraction algorithm to solve the approximation problem. In the end, the method is applied to some numerical experiments and the effectiveness of the algorithm is verified.
基金Project supported by the National Natural Science Foundation of China(Nos.11790282,12172235,12072208,and 52072249)the Opening Foundation of State Key Laboratory of Shijiazhuang Tiedao University of China(No.ZZ2021-13)。
文摘A stochastic wheelset model with a nonlinear wheel-rail contact relationship is established to investigate the stochastic stability and stochastic bifurcation of the wheelset system with the consideration of the stochastic parametric excitations of equivalent conicity and suspension stiffness.The wheelset is systematized into a onedimensional(1D)diffusion process by using the stochastic average method,the behavior of the singular boundary is analyzed to determine the hunting stability condition of the wheelset system,and the critical speed of stochastic bifurcation is obtained.The stationary probability density and joint probability density are derived theoretically.Based on the topological structure change of the probability density function,the stochastic Hopf bifurcation form and bifurcation condition of the wheelset system are determined.The effects of stochastic factors on the hunting stability and bifurcation characteristics are analyzed,and the simulation results verify the correctness of the theoretical analysis.The results reveal that the boundary behavior of the diffusion process determines the hunting stability of the stochastic wheelset system,and the left boundary characteristic value cL=1 is the critical state of hunting stability.Besides,stochastic D-bifurcation and P-bifurcation will appear in the wheelset system,and the critical speeds of the two kinds of stochastic bifurcation decrease with the increase in the stochastic parametric excitation intensity.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11902081)the Science and Technology Projects of Guangzhou (Grant No. 202201010326)the Guangdong Provincial Basic and Applied Basic Research Foundation (Grant No. 2023A1515010833)。
文摘This paper focuses on the stochastic analysis of a viscoelastic bistable energy harvesting system under colored noise and harmonic excitation, and adopts the time-delayed feedback control to improve its harvesting efficiency. Firstly, to obtain the dimensionless governing equation of the system, the original bistable system is approximated as a system without viscoelastic term by using the stochastic averaging method of energy envelope, and then is further decoupled to derive an equivalent system. The credibility of the proposed method is validated by contrasting the consistency between the numerical and the analytical results of the equivalent system under different noise conditions. The influence of system parameters on average output power is analyzed, and the control effect of the time-delayed feedback control on system performance is compared. The output performance of the system is improved with the occurrence of stochastic resonance(SR). Therefore, the signal-to-noise ratio expression for measuring SR is derived, and the dependence of its SR behavior on different parameters is explored.
基金supported by the National Natural Science Foundation of China(10902094,10932009,11072212 and 11272279)the Special Foundation for Young Scientists of Fujian Province of China(2008F3100)
文摘The approximate transient response of quasi in- tegrable Hamiltonian systems under Gaussian white noise excitations is investigated. First, the averaged It6 equa- tions for independent motion integrals and the associated Fokker-Planck-Kolmogorov (FPK) equation governing the transient probability density of independent motion integrals of the system are derived by applying the stochastic averag- ing method for quasi integrable Hamiltonian systems. Then, approximate solution of the transient probability density of independent motion integrals is obtained by applying the Galerkin method to solve the FPK equation. The approxi- mate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coeffi- cients. The transient probability densities of displacements and velocities can be derived from that of independent mo- tion integrals. Three examples are given to illustrate the ap- plication of the proposed procedure. It is shown that the re- suits for the three examples obtained by using the proposed procedure agree well with those from Monte Carlo simula- tion of the original systems.
基金supported by the National Natural Science Foundation of China(10932009,11072212,11272279,and 11002059)the Specialized Research Fund for the Doctoral Program of Higher Education(20103501120003)the Fundamental Research Funds for Huaqiao University(JB-SJ1010)
文摘We studied the response of fractional-order van de Pol oscillator to Gaussian white noise excitation in this letter. An equivalent integral-order nonlinear stochastic system is obtained to replace the given system based on the principle of minimum mean-square error. Through stochastic averaging, an averaged Ito equation is deduced. We obtained the Fokker–Planck–Kolmogorov equation connected to the averaged Ito equation and solved it to yield the approximate stationary response of the system. The analytical solution is confirmed by using Monte Carlo simulation.