Fokker Planck Kolmogorov Research Articles

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.

Performance analysis of nonlinear vibration energy harvesting system with inelastic barrier under colored noise excitation

The vibro-impact structures are introduced into the piezoelectric vibration energy harvesting (VEH) systems, which can broaden the operating bandwidth of the piezoelectric VEH systems and thus improve the harvesting efficiency. The process of vibro-impact not only causes energy loss during the impact process but also causes the local deformation between two impact bodies. Therefore, this manuscript proposes a class of stochastic non-classical inelastic impact VEH models, whose non-classical impact processes are described by Hertz contact forces. Firstly, the relationship between the total energy of the unperturbed system and the potential energy of the system at the barrier position is used to determine whether the impact phenomena have occurred or not, then used to derive the period and frequency of the system. Secondly, using the quasi-conservative stochastic averaging method, the averaged drift term and the averaged diffusion term of the piecewise smooth averaged Ito^ equation are derived. The stationary probability density functions (PDFs), the mean square output (MSO) voltage and the root mean square (RMS) voltage of the system response can be obtained from solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation for averaged Ito^ equation. Finally, an example is provided to verify the effectiveness of the proposed method, and the effects of the model parameters on the MSO voltage and power conversion efficiency (PCE) are analyzed. The relationship between the MSO voltage and the PCE is also analyzed. It is of great significance for the design and optimization of stochastic VEH systems.

Mean Field Game-based Decentralized Optimal Charging Control for large-scale of electric Vehicles

In this paper, the online decentralized optimal charging control problem for large-scale electric vehicles is investigated. Massive concurrent charging, which often happens for the electric buses in a smart city, is devastating to the power grid's supply capacity due to the instant high load demand. Traditional strategies rely on a centralized dispatch center which suffers from the rising computational complexity and the data communication burden when the bus number is huge. In the novel design, a decentralized optimal charging control strategy is developed to tackle these challenges. By engaging the emerging Mean Field Games (MFG) theory with a novel Actor-Critic-Mass (ACM) reinforcement learning scheme, a probability density function (PDF) of the massive state of charge (SOC) is used to encode all electric vehicles’ charging demand information into a fix-dimension function. Moreover, the optimal charging control and the PDF of massive SOC can be solved through the coupled Fokker-Planck-Kolmogorov (FPK) and Hamilton-Jacobi-Bellman (HJB) equations in a decentralized manner. Eventually, the effectiveness of the proposed scheme is validated through a series of numerical simulations.

Determination of monopile offshore structure response to stochastic wave loads via analog filter approximation and GV-GDEE procedure

Efficient and accurate stochastic response analysis of offshore structures subjected to ocean wave loads has been a challenging problem for several decades. This is especially true for multi-degree-of-freedom nonlinear structures when information about the probability density is desired. How to account for the random field of the wave loads, and how to deal with the coupling of high-dimensional partial differential equation governing the joint probability density functions are among the major challenges. To this end, in the present paper an approach is proposed by incorporating the filter approximation of random fields of sea waves with the recently developed Globally-evolving based generalized density evolution equation (GV-GDEE). To deal with the first issue regarding the random wave filed, the analog filter technique is adopted. In particular, a filter approximation of the wave kinematics field distributed over a vertical line along the ocean domain is proposed. It comprises a linear differential equation set with white noise input. An augmented system is thereby constructed by incorporating the additional differential equation set yielding wave excitation simulated by an analog filter into the equation of motion of a monopile offshore structure. To circumvent the issue of coupling of high-dimensional equations for this augmented system with even higher dimension, the GV-GDEE approach is employed. Specifically, by constructing the effective drift coefficients based on the high-dimensional drift coefficients in the associated Fokker-Planck-Kolmogorov (FPK) equation, a two-dimensional GV-GDEE in terms of the response quantity of interest is derived. Further, this GV-GDEE is solved by the path integral method. Furthermore, two numerical examples are included. In particular, the nonstationary stochastic responses of a 5 megawatt (MW) fixed monopile offshore wind turbine structure under two operating conditions are studied. Comparisons with pertinent Monte Carlo simulations (MCS) demonstrate the accuracy and efficiency of the proposed approach.

Variance and Entropy Assignment for Continuous-Time Stochastic Nonlinear Systems.

This paper investigates the randomness assignment problem for a class of continuous-time stochastic nonlinear systems, where variance and entropy are employed to describe the investigated systems. In particular, the system model is formulated by a stochastic differential equation. Due to the nonlinearities of the systems, the probability density functions of the system state and system output cannot be characterised as Gaussian even if the system is subjected to Brownian motion. To deal with the non-Gaussian randomness, we present a novel backstepping-based design approach to convert the stochastic nonlinear system to a linear stochastic process, thus the variance and entropy of the system variables can be formulated analytically by the solving Fokker–Planck–Kolmogorov equation. In this way, the design parameter of the backstepping procedure can be then obtained to achieve the variance and entropy assignment. In addition, the stability of the proposed design scheme can be guaranteed and the multi-variate case is also discussed. In order to validate the design approach, the simulation results are provided to show the effectiveness of the proposed algorithm.

Hybrid stochastic differential systems in pharmacokinetics

AbstractModelling patient medication poor compliance combined with the inherent uncertainty in a bioavailability trial leads recognizably to the fact that plasma concentration-time curve must be treated as a random process. We suggest for modelling a general class of hybrid stochastic differential systems (HSDS). Roughly speaking, a HSDS is a piecewise diffusion process with jumps of two types: spontaneous and predictable. The essential tasks for us afterwards are to identify the model ingredients and to derive its bearings; of interest are a couple of operators associated to every Markov process: the generating operator and the Fokker–Planck operator. The model induces a Fokker–Planck–Kolmogorov equation along with moment equations, and computations based on direct solutions of the latter make it possible to study the variability of the concentration around its mean as compared to the full compliance case and to assess the effect of some parameters such as the intake and elimination rates.

Response and reliability analysis of random time-delay controlled systems undergo wide-band random excitations

This paper proposed a new procedure for response and reliability analysis of a multi-degree-of-freedom (MDOF) random time-delay controlled nonlinear system subjected to wide-band random excitations. A two-steps method is presented to deal with random time delay: Firstly, by modeling the random jump time-delay as a Markov jump process, the random delayed control is formulated as the one with discrete Markov jump varying delay; secondly, based on the randomly periodic property of the system, the Markov jump delay is taken out of the control as a parameter. A new scheme based on limit averaging and stochastic averaging principle is developed, which suppresses the rapidly varying processes to generate a less-dimensional probability-weighted system. The Fokker–Planck–Kolmogorov (FPK) equation and the backward Kolmogorov (BK) equation are then deduced and solved, from which the quantitative characteristics of the response and reliability of the system are obtained. Finally, an example is worked out in detail, from which the application and effectiveness of the proposed method are demonstrated.

Developing an innovative method to control the probability density function shape of the state response for nonlinear stochastic systems

AbstractFor nonlinear stochastic systems, controlling low‐order moment indexes, such as mean and variance, may not effectively achieve actual control requirements since they do not reflect the complete statistical characteristics of random variables. Instead, the probability density function (PDF), which provides complete characterization of statistical properties, can be used to control each order moment more efficiently. Existing PDF shape control methods are mainly suited for nonlinear systems with polynomial forms, which leads to limitations in their applications. In this article, we propose a PDF shape control method using the Fokker–Planck–Kolmogorov (FPK) equation for nonlinear stochastic systems with arbitrary expression. In the proposed method, the exact stationary solution of the FPK equation is derived first. If the nonlinear function of the stochastic system is not polynomial, it is first converted into a polynomial form using the approximation method. The controller is then devised for PDF shape control, and the parameters of the controller are optimized to obtain the optimal controller. We conducted simulation experiments to validate the effectiveness of the proposed method. The results show that the proposed method can effectively control the PDF shape of the state response for nonlinear stochastic systems, regardless of whether the target PDF is unimodal, symmetric bimodal, or asymmetric bimodal. Compared with the Fourier transform method, the control effect of the proposed method is much better. The results suggest that the proposed method provides a scientific basis for wider applications of the PDF shape control method.

On the construction of a family of anomalous-diffusion Fokker–Planck−Kolmogorov’s equations based on the Sharma–Taneja–Mittal entropy functional

A logical scheme for constructing thermodynamics of anomalous stochastic systems based on the nonextensive two-parameter (κ, ς) -entropy of Sharma–Taneja–Mittal (SHTM) is considered. Thermodynamics within the framework (2 - q) -statistics of Tsallis was constructed, which belongs to the STM family of statistics. The approach of linear nonequilibrium thermodynamics to the construction of a family of nonlinear equations of Fokker−Planck−Kolmogorov (FPK), is used, correlated with the entropy of the STM, in which the stationary solution of the diffusion equation coincides with the corresponding generalized Gibbs distribution obtained from the extremality (κ, ς) - entropy condition of a non-additive stochastic system. Taking into account the convexity property of the Bregman divergence, it was shown that the principle of maximum equilibrium entropy is valid for (κ, ς) - systems, and also was proved the H - theorem determining the direction of the time evolution of the non-equilibrium state of the system. This result is extended also to non-equilibrium systems that evolve to a stationary state in accordance with the nonlinear FPK equation. The method of the ansatz- approach for solving non-stationary FPK equations is considered, which allows us to find the time dependence of the probability density distribution function for non-equilibrium anomalous systems. Received diffusive equations FPК can be used, in particular, at the analysis of diffusion of every possible epidemics and pandemics. The obtained diffusion equations of the FPK can be used, in particular, in the analysis of the spread of various epidemics and pandemics.

ON THE LAW OF DISTRIBUTION OF THE RATE OF FATIGUE FRACTURE

The article discusses the correction of the law of development of fatigue cracks based on the application of the Fokker–Planck–Kolmogorov equation. The coefficient of variation of the fatigue crack propagation rate was chosen as the correction criterion. Its decrease brings the descriptions of the fatigue fracture process closer to engineering. The latter is especially important for specialists dealing with the assessment of the fatigue strength of metal-consuming structures, where mass formation of fatigue cracks is inevitable.

Модель процесса разделения суспензий в гидродинамических фильтрах с вращающимся фильтрующим элементом

The possibility of applying the probabilistic-statistical method for theoretical calculation of characteristics related to suspensions separation in self-cleaning hydrodynamic filters with a rotating filter element has been demonstrated. It has been shown that the change in characteristics of the suspensions separation in hydrodynamic filters with a rotating filter element can be satisfactorily described based on equations of diffusion type, in particular, using the Fokker–Planck–Kolmogorov equation. The main parameters of the separation process in hydrodynamic filters with a rotating filter element have been determined.

Acceleration spectrum analysis of hyperbolic tangent package under random excitation

AbstractMany packaging cushion materials follow the hyperbolic tangent type of constitutive relation, so a hyperbolic tangent stiffness non‐linearity was derived to describe the stiffness property of the package. Transport package was modelled as the single degree of freedom (SDOF) of hyperbolic tangent spring–mass–damping (SMD) system. The displacement and velocity responses of the system were obtained by Fokker–Planck–Kolmogorov (FPK) equation. Based on the kinematic equation and integral transforming, the approximate analytic solution of the acceleration response power spectral density (PSD) of the hyperbolic tangent package was established for the first time, and the influences of system non‐linearities on the acceleration response PSD were investigated. Validation experiment with a complete packaged computer was carried out to verify the proposed approach. Non‐linearity of the stiffness influenced the acceleration response PSD through the effect of parameter γ and statistical moment mi0. The frequency corresponding to acceleration response peak was determined by the damping correlation coefficient α and parameter γ, which was determined by system characteristic parameter β, frequency parameter ω0 and statistical moment mi0. The proposed approach can be used to predict the acceleration response PSD of transport package, and the analysis has important value for packaging optimization design.

Resonance response analysis of nonlinear vibration energy harvesting system under bounded noise excitation

In this paper, the resonance response of piezoelectric vibration energy harvester (VEH) driven by bounded noise is discussed through the quasi-conservative stochastic averaging method. A nonlinear transformation based on the total energy is firstly established to transform piezoelectric VEH system from an electromechanical coupled nonlinear system into a single-degree-of-freedom (SDOF) system. Then the SDOF system is rewritten as It\^o stochastic system about the energy and residual phase under the case of p:q resonance through the quasi-conservative stochastic averaging method. And the joint probability density function (JPDF) of the stationary response is obtained by solving the corresponding two-dimensional Fokker-Planck-Kolmogorov (FPK) equation using the finite difference method. Meanwhile, the mean-square electric voltage and the mean output power are further analytically given through the JPDF. Finally, the resonance response of piezoelectric VEH system is analyzed in detail in case of the primary resonance, and the Monte Carlo (MC) simulation technique is adopted to validate the effectiveness of the finite difference method.

Study on Wear Dynamic Reliability of Gear System Based on Markov Diffusive Process

Abstract Wear is one of the most common failure forms of gear transmission system. In this paper, a dynamic wear reliability evaluation method for gear system, which subjects to stochastic external load, is proposed on the basis of the Markov diffusive process. The stochastic load is considered as the combination of constant load and random noise. The failure is defined as the wear depth of the maximum wear point of gear under the constant load that exceeds a specific threshold. The maximum wear point and relative sliding velocity are obtained by deterministic wear analysis. The stochastic noise is assumed as Gaussian white noise; hence, the wear depth can be described as Markov diffusive process, and the transition probability is governed by Fokker–Planck–Kolmogorov (FPK) equation. With the transition probability function, the wear life and dynamic reliability of gear systems with different noise spectral densities are predicted. The results reveal that the wear depth obeys normal distribution and becomes more and more scattered with noise spectral density increases. The wear life does not obey normal distribution, and the effect of noise spectral density on mean life is neglectable while on a standard deviation of life is considerable.

On Nonuniqueness of Probability Solutions to the Cauchy Problem for the Fokker–Planck–Kolmogorov Equation

On Nonuniqueness of Probability Solutions to the Cauchy Problem for the Fokker–Planck–Kolmogorov Equation

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.

McKean–Vlasov SDEs under measure dependent Lyapunov conditions

We prove the existence of weak solutions to McKean–Vlasov SDEs defined on a domain D⊆Rd with continuous and unbounded coefficients and degenerate diffusion coefficient. Using differential calculus for the flow of probability measures due to Lions, we introduce a novel integrated condition for Lyapunov functions in an infinite dimensional space D×P(D), where P(D) is a space of probability measures on D. Consequently we show existence of solutions to the McKean–Vlasov SDEs on [0,∞). This leads to a probabilistic proof of the existence of a stationary solution to the nonlinear Fokker–Planck–Kolmogorov equation under very general conditions. Finally, we prove uniqueness under an integrated condition based on a Lyapunov function. This extends the standard monotone-type condition for uniqueness.

Random vibration of pretensioned rectangular membrane structures under heavy rainfall excitation

This paper analytically and experimentally investigates the random vibration problem of pretensioned rectangular membrane structures under heavy rainfall. An analytical model is firstly proposed. In this model, the heavy rainfall is modeled as a series of discretized pulse load whose amplitude follows the non-Gaussian distribution; then, the nonlinear governing equation and corresponding Fokker Planck Kolmogorov (FPK) equation of membrane structures are derived and solved by the perturbation method, considering both the geometrical nonlinearity of structure and the non-Gaussian type of load; consequently, the analytical solutions of the probability density function (PDF), mean and standard deviation of displacement response can be obtained. An artificial heavy rainfall experimental system is then developed. This system is used to (i) verify the analytical model and (ii) evaluate the effects of rainfall intensity, pretension level, and material properties. The results show that the displacement follows the significant non-Gaussian distribution and proves strong nonlinear characteristics with increasing rainfall intensity and pretension. The proposed analytical model can guide engineers to design membrane structures from the perspective of probability.

Rate-dependent bifurcation dodging in a thermoacoustic system driven by colored noise

Tipping in multistable systems occurs usually by varying the input slightly, resulting in the output switching to an often unsatisfactory state. This phenomenon is manifested in thermoacoustic systems. The thermoacoustic instability may lead to the disintegration of rocket engines, gas turbines and aeroengines, so it is necessary to design control measures for its suppression. It was speculated that such unwanted instability states may be dodged by changing the bifurcation parameters quickly enough, and compared with the white noise discussed in [1], colored noise with nonzero correlation time is more practical and important to the system. Thus, in this work, based on a fundamental mathematical model of thermoacoustic systems driven by colored noise, the corresponding Fokker–Planck–Kolmogorov equation of the amplitude is derived by using a stochastic averaging method. A transient dynamical behavior is identified through a probability density analysis. We find that both a relatively higher rate of change of parameters and change in the correlation time of the noise are beneficial to dodge thermoacoustic instability, while a relatively large noise intensity is a disadvantageous factor. More interestingly and importantly, power-law relationships between the maximum amplitude and the noise parameters are uncovered, and the probability of successfully dodging a thermoacoustic instability is calculated. These results serve as a guidance for the design of engines and to propose an effective control strategy, which is of great significance to aerospace-related fields.

A generalization of the Lagrange–Hamilton formalism with application to non-conservative systems and the quantum to classical transition

This work has two aims. The first is to develop a Lagrange–Hamilton framework for the analysis of multi-degree-of-freedom nonlinear systems in which non-conservative effects are included in the variational principle of least action from the outset. The framework is a generalization of the Bateman approach in which a set of adjoint coordinates is introduced. A function termed the M-function is introduced as the Fourier transform over the momenta of the joint probability density function (JPDF) of the displacements and momenta, and it is shown that for statistical systems, this function can be written as an expectation involving the new principle function and a general dimensional constant ℏ. This leads to a concise derivation of the Fokker–Planck–Kolmogorov equation. It is found that the equation governing the M-function can be expressed in terms of the new Hamiltonian by replacing momenta by differential operators, meaning that the function satisfies the same equation as the quantum wave function. This gives rise to the second aim of this work: to explore relations between the developed classical framework and quantum mechanics. It is shown that for an undamped linear system, the solution of the M-function equation yields the response JPDF as a sum of Wigner functions. This classical analysis leads to a number of well-known results from quantum mechanics as ℏ → 0, and the extension of this result to nonlinear systems is discussed. The quantum wave function associated with the Hamiltonian is then considered, and the relevance of this function to the physical system is discussed.