Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order α

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
White Noise ◽  
Fractional Order ◽  
Gaussian White Noise ◽  
Linear Partial Differential Equation ◽  
Planck Equation ◽  
Stochastic Dynamical Systems ◽  
Poissonian Noise ◽  
Liouville Fractional Derivative ◽  
White Noises ◽  
Taylor’S Series

AbstractIn a first stage, the paper deals with the derivation and the solution of the equation of the probability density function of a stochastic system driven simultaneously by a fractional Gaussian white noise and a fractional Poissonian white noise both of the same order. The key is the Taylor’s series of fractional order f(x + h) = E α(hαDx α)f(x) where E α() denotes the Mittag-Leffler function, and D x α is the so-called modified Riemann-Liouville fractional derivative which removes the effects of the non-zero initial value of the function under consideration. The corresponding fractional linear partial differential equation is solved by using a suitable extension of the Lagrange’s technique involving an auxiliary set of fractional differential equations. As an example, one considers a half-oscillator of fractional order driven by a fractional Poissonian noise.

scholarly journals Calculus on Gaussian and Poisson white noises

1988 ◽  
Vol 111 ◽  
pp. 41-84 ◽  
Author(s):  
Yoshifusa Ito ◽  
Izumi Kubo
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Exceptional Point ◽  
Stochastic Integrals ◽  
Generalized Poisson ◽  
Gaussian Case ◽  
White Noises

Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc. ([8], [9]), analogously to the works of T. Hida ([3], [4], [5]). Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf. [10], [11], [12], [13]). Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc. in a way completely parallel with the Gaussian case. The only one exceptional point, which is most significant, is that the multiplications are described by for the Gaussian case, for the Poisson case,as will be stated in Section 5. Conversely, those formulae characterize the types of white noises.

Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller

2018 ◽  
Vol 18 (06) ◽  
pp. 1850083 ◽  
Author(s):  
Wei Li ◽  
Lincong Chen ◽  
Junfeng Zhao ◽  
Natasa Trisovic
Keyword(s):  
Dynamical Systems ◽  
Fractional Order ◽  
First Passage Time ◽  
Gaussian White Noise ◽  
Passage Time ◽  
Reliability Estimation ◽  
Kolmogorov Equation ◽  
Critical Parameters ◽  
Stochastic Dynamical Systems ◽  
Stochastic Dynamical System

In this paper, the reliability of stochastic dynamical systems under Gaussian white noise excitations with fractional order proportional–inegral–derivative (FOPID) controller is estimated. First, the FOPID controller is approximated by a set of combination of displacement and velocity based on the generalized van der Pol transformation. Then, the stochastic averaging method of energy envelope is applied to obtain a diffusive differential equation, from which the Backward Kolmogorov equation, governing the conditional reliability function, and the Generalized Pontryagin equation, governing the statistical moments of first-passage time, are derived from the averaged equation and solved numerically. Finally, in the two examples, the critical parameters in the FOPID controller are shown to be capable of improving the reliability of the stochastic dynamical system apparently, and all numerical results are verified to be efficient and correct by the Monte Carlo simulation.

Noise Influenced Responses of Elastic Cantilevers With Nonlinear Tip Force Interactions

2011 ◽  
Author(s):  
Ishita Chakraborty ◽  
Balakumar Balachandran
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Harmonic Excitation ◽  
Attractive Force ◽  
Surface Band ◽  
Micro Scale ◽  
Cantilever Structure ◽  
Macro Scale ◽  
Band Limited

In this article, the effects of noise on a base-excited cantilever structure with nonlinear tip force interactions are studied by using experimental, numerical, and analytical methods. The focus of the study is on the enhancement of the cantilever response, when Gaussian white noise is added to the harmonic base input. The experimental arrangement consists of a base-excited elastic cantilever with a magnet attached to its free end. An attractive force is generated at the cantilever tip magnet through another magnet of opposite polarity, which is fixed to a translatory stage. The second magnet is covered by a thin compliant material, with which the tip magnet makes intermittent contact when the cantilever is subjected to a base excitation. For a purely harmonic excitation, it is observed that the tip magnet of the cantilever sticks to the base magnet due to the attractive force. Starting from a situation where the cantilever tip is sticking to the surface, band-limited white Gaussian noise is added to the excitation and the strength of noise is gradually increased. The cantilever tip resumes its periodic motion when the strength of added noise reaches a sufficient signal to noise ratio. This phenomenon is explored by using a reduced-order numerical model and an analytical framework involving the application of a moment-evolution approximation derived from the associated Fokker Planck equation for the system. Since the macro-scale experimental system qualitatively replicates the micro-scale attractive-repulsive force interaction experienced by an atomic force microscope cantilever operated in tapping mode, this study sheds light on the possible use of white noise to control the sticking of such micro-scale cantilevers with sample surfaces.

scholarly journals GENERALIZED WIENER PROCESS AND KOLMOGOROV'S EQUATION FOR DIFFUSION INDUCED BY NON-GAUSSIAN NOISE SOURCE

2005 ◽  
Vol 05 (02) ◽  
pp. L267-L274 ◽  
Author(s):  
ALEXANDER DUBKOV ◽  
BERNARDO SPAGNOLO
Keyword(s):  
White Noise ◽  
Wiener Process ◽  
Anomalous Diffusion ◽  
Gaussian Noise ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Non Gaussian ◽  
Generalized Wiener Process ◽  
Fokker Planck

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker–Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov–Feller equation for discontinuous Markovian processes, and the fractional Fokker–Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.

Probabilistic Solution of Vibro-Impact Systems Under Additive Gaussian White Noise

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
H. T. Zhu
Keyword(s):  
White Noise ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Solution Procedure ◽  
Additive Gaussian White Noise ◽  
Stationary Probability Density Function ◽  
Probabilistic Solution ◽  
Zero Offset ◽  
Polynomial Closure

This paper presents a solution procedure for the stationary probability density function (PDF) of the response of vibro-impact systems under additive Gaussian white noise. The constraint is a unilateral zero-offset barrier. The vibro-impact system is first converted into a system without barriers using the Zhuravlev nonsmooth coordinate transformation. The stationary PDF of the converted system is governed by the Fokker–Planck equation which is solved by the exponential-polynomial closure (EPC) method. A vibro-impact Duffing oscillator with either elastic or lightly inelastic impacts is considered in a numerical analysis. Meanwhile, the level of nonlinearity in displacement is also examined in this study as well as the case of negative linear stiffness. Comparison with the simulated results shows that the EPC method can present a satisfactory PDF for displacement and velocity when the polynomial order is taken as 4 in the investigated cases. The tail of the PDF also works well with the simulated result.

scholarly journals On stochastic differential equations driven by the renormalized square of the Gaussian white noise

2015 ◽  
Vol 18 (04) ◽  
pp. 1550025
Author(s):  
Bilel Kacem Ben Ammou ◽  
Alberto Lanconelli
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Gaussian White Noise ◽  
Wick Product ◽  
Nonlinear Functions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Definition Of ◽  
White Noises

We investigate the properties of the Wick square of Gaussian white noises through a new method to perform nonlinear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove an Itô-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, global Lipschitz continuity, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.

A Wavelet Spatial Correlation Algorithm to Partial Discharge Denoising Based on MDL Criterion

2013 ◽  
Vol 340 ◽  
pp. 642-646
Author(s):  
Li Song Tian ◽  
Wei Xuan Chen
Keyword(s):  
White Noise ◽  
Spatial Correlation ◽  
Partial Discharge ◽  
Parameters Estimation ◽  
Optimization Strategy ◽  
Signal Parameter ◽  
Detection Systems ◽  
Correlation Algorithm ◽  
White Noises ◽  
General Method

The partial discharge (PD) detection systems are often vulnerable to strong external interferences, and sometimes the PD signals are submerged in noises (white noise for example) completely. So the signals acquired must be preprocessed to obtain the reliable PD information. While there are many methods for white noise denoising, mostly are not very suitable for partial discharge. The wavelet transform (WT) coefficient of PD and white noises have different spread characteristics in different WT scales. Based on the Information Theory, The Minimum Information Description Length (MDL) criterion is a optimization strategy, a small amount of signal parameter is requried to the PD signals representation, the paper proposes a wavelet spatial correlation algorithm to partial discharge denoising based on MDL criterion: optimal wavelet function is selected based on MDL, then have the white noise reduced in WT, the algorithm has wonderful virtues such as free from any parameters estimation about noise, free from presetting threshhold and threshold chooseing behavior, so the algorithm is highly adaptive. Large amount of experimental results illustrate that the method presented in this paper are efficient and feasible and outperforms other general method of PD noise reduction.

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