Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems

1997 ◽  
Vol 64 (1) ◽  
pp. 157-164 ◽  
Author(s):  
W. Q. Zhu ◽  
Y. Q. Yang
Keyword(s):  
Nonlinear System ◽  
Probability Density ◽  
Hamiltonian Systems ◽  
Present Method ◽  
Transition Probability ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Kolmogorov Equation ◽  
Transition Probability Density ◽  
System Method

A stochastic averaging method is proposed to predict approximately the response of multi-degree-of-freedom quasi-nonintegrable-Hamiltonian systems (nonintegrable Hamiltonian systems with lightly linear and (or) nonlinear dampings and subject to weakly external and (or) parametric excitations of Gaussian white noises). According to the present method, a one-dimensional approximate Fokker-Planck-Kolmogorov equation for the transition probability density of the Hamiltonian can be constructed and the probability density and statistics of the stationary response of the system can be readily obtained. The method is compared with the equivalent nonlinear system method for stochastically excited and dissipated nonintegrable Hamiltonian systems and extended to a more general class of systems. An example is given to illustrate the application and validity of the present method and the consistency of the present method and the equivalent nonlinear system method.

Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems Under Poisson White Noise Excitation

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Y. Zeng ◽  
W. Q. Zhu
Keyword(s):  
Probability Density ◽  
Hamiltonian Systems ◽  
Transition Probability ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Degree Of Freedom ◽  
Kolmogorov Equation ◽  
Transition Probability Density ◽  
Poisson White Noise ◽  
Van Der Pol Oscillators

A stochastic averaging method for predicting the response of multi-degree-of-freedom quasi-nonintegrable-Hamiltonian systems (nonintegrable-Hamiltonian systems with lightly linear and (or) nonlinear dampings subject to weakly external and (or) parametric excitations of Poisson white noises) is proposed. A one-dimensional averaged generalized Fokker–Planck–Kolmogorov equation for the transition probability density of the Hamiltonian is derived and the probability density of the stationary response of the system is obtained by using the perturbation method. Two examples, two linearly and nonlinearly coupled van der Pol oscillators and two-degree-of-freedom vibro-impact system, are given to illustrate the application and validity of the proposed method.

Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations

2013 ◽  
Vol 81 (4) ◽  
Author(s):  
Wantao Jia ◽  
Weiqiu Zhu ◽  
Yong Xu ◽  
Weiyan Liu
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Hamiltonian Systems ◽  
Transition Probability ◽  
Digital Simulation ◽  
Relaxation Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Transition Probability Density ◽  
Poisson White Noise

A stochastic averaging method for quasi-integrable and resonant Hamiltonian systems subject to combined Gaussian and Poisson white noise excitations is proposed. The case of resonance with α resonant relations is considered. An (n + α)-dimensional averaged Generalized Fokker–Plank–Kolmogorov (GFPK) equation for the transition probability density of n action variables and α combinations of phase angles is derived from the stochastic integrodifferential equations (SIDEs) of original quasi-integrable and resonant Hamiltonian systems by using the jump-diffusion chain rule. The reduced GFPK equation is solved by using finite difference method and the successive over relaxation method to obtain the stationary probability density of the system. An example of two nonlinearly damped oscillators under combined Gaussian and Poisson white noise excitations is given to illustrate the proposed method. The good agreement between the analytical results and those from digital simulation shows the validity of the proposed method.

Stochastic Averaging of Quasi-Integrable Hamiltonian Systems

1997 ◽  
Vol 64 (4) ◽  
pp. 975-984 ◽  
Author(s):  
W. Q. Zhu ◽  
Z. L. Huang ◽  
Y. Q. Yang
Keyword(s):  
Probability Density ◽  
Hamiltonian Systems ◽  
Transition Probability ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Integrable Hamiltonian Systems ◽  
Transition Probability Density ◽  
Special Cases ◽  
Fpk Equation ◽  
Action Variables

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems, i.e., multi-degree-of-freedom integrable Hamiltonian systems subject to lightly linear and (or) nonlinear dampings and weakly external and (or) parametric excitations of Gaussian white noises. According to the present method an n-dimensional averaged Fokker-Planck-Kolmogrov (FPK) equation governing the transition probability density of n action variables or n independent integrals of motion can be constructed in nonresonant case. In a resonant case with α resonant relations, an (n + α)-dimensional averaged FPK equation governing the transition probability density of n action variables and α combinations of phase angles can be obtained. The procedures for obtaining the stationary solutions of the averaged FPK equations for both resonant and nonresonant cases are presented. It is pointed out that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are two special cases of the present stochastic averaging. Two examples are given to illustrate the application and validity of the proposed method.

The stationary moments of Poisson-driven non-linear dynamical systems

1982 ◽  
Vol 19 (3) ◽  
pp. 702-706
Author(s):  
Charles E. Smith ◽  
Loren Cobb
Keyword(s):  
Dynamical Systems ◽  
Probability Density ◽  
Stochastic Systems ◽  
Transition Probability ◽  
Kolmogorov Equation ◽  
Transition Probability Density ◽  
Linear Dynamical Systems ◽  
Stationary Density ◽  
Non Linear ◽  
Noise Input

Moment recursion relations have previously been derived for the stationary probability density functions of continuous-time stochastic systems with Wiener (white noise) input. These results are extended in this paper to the case of Poisson (shot noise) input. The non-linear dynamical systems are expressed, in general, as stochastic differential equations, with an independent increment input. The transition probability density function evolves according to the appropriate Kolmogorov equation. Moments of the stationary density are obtained from the Fourier transform of the stationary density. The moment relations can be used to estimate the parameters of linear and non-linear stochastic systems from empirical moments, given either Wiener or Poisson input.

Equivalent Nonlinear System Method for Stochastically Excited Hamiltonian Systems

1994 ◽  
Vol 61 (3) ◽  
pp. 618-623 ◽  
Author(s):  
W. Q. Zhu ◽  
T. T. Soong ◽  
Y. Lei
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
White Noise ◽  
Probability Density ◽  
Hamiltonian Systems ◽  
Gaussian White Noise ◽  
Energy Balancing ◽  
Dissipation Energy ◽  
System Method ◽  
Approximate Probability

An equivalent nonlinear system method is presented to obtain the approximate probability density for the stationary response of multi-degree-of-freedom nonlinear Hamiltonian systems to Gaussian white noise parametric and/or external excitations. The equivalent nonlinear systems are obtained on the basis of one of the following three criteria: least mean-squared deficiency of damping forces, dissipation energy balancing, and least mean-squared deficiency of dissipation energies. An example is given to illustrate the application and validity of the method and the differences in the three equivalence criteria.

The stationary moments of Poisson-driven non-linear dynamical systems

1982 ◽  
Vol 19 (03) ◽  
pp. 702-706
Author(s):  
Charles E. Smith ◽  
Loren Cobb
Keyword(s):  
Dynamical Systems ◽  
Probability Density ◽  
Stochastic Systems ◽  
Transition Probability ◽  
Kolmogorov Equation ◽  
Transition Probability Density ◽  
Linear Dynamical Systems ◽  
Stationary Density ◽  
Non Linear ◽  
Noise Input

Moment recursion relations have previously been derived for the stationary probability density functions of continuous-time stochastic systems with Wiener (white noise) input. These results are extended in this paper to the case of Poisson (shot noise) input. The non-linear dynamical systems are expressed, in general, as stochastic differential equations, with an independent increment input. The transition probability density function evolves according to the appropriate Kolmogorov equation. Moments of the stationary density are obtained from the Fourier transform of the stationary density. The moment relations can be used to estimate the parameters of linear and non-linear stochastic systems from empirical moments, given either Wiener or Poisson input.

scholarly journals Parametrix construction for certain Lévy-type processes

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Victoria Knopova ◽  
Alexei Kulik
Keyword(s):  
Markov Process ◽  
Probability Density ◽  
Lower Bounds ◽  
Transition Probability ◽  
Local Operator ◽  
Upper And Lower Bounds ◽  
Transition Probability Density ◽  
Strong Markov Process ◽  
Non Local ◽  
Strong Markov

AbstractIn this paper, we show that a non-local operator of certain type extends to the generator of a strong Markov process, admitting the transition probability density. For this transition probability density we construct the intrinsic upper and lower bounds, and prove some smoothness properties. Some examples are provided.

An improved energy envelope stochastic averaging method and its application to a nonlinear oscillator

2016 ◽  
Vol 23 (1) ◽  
pp. 119-130 ◽  
Author(s):  
Yaping Zhao
Keyword(s):  
Steady State ◽  
Probability Density Function ◽  
Probability Density ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
System Response ◽  
Mean Square ◽  
Fpk Equation ◽  
The Mean

An improved stochastic averaging method of the energy envelope is proposed, whose application sphere is extensive and whose implementation is convenient. An oscillating system with both nonlinear damping and stiffness is taken into account. Its averaged Fokker-Planck-Kolmogorov (FPK) equation in respect of the transition probability density function of the energy envelope is deduced by virtue of the method mentioned above. Under the initial and boundary conditions, the joint probability density function as to the displacement and velocity of the system is worked out in closed form after solving the averaged FPK equation by right of a technique based on the integral transformation. With the aid of the special functions, the transient solutions of the probabilistic characteristics of the system response are further derived analytically, including the probability density functions and the mean square values. A simple approach to generate the ideal white noise is drastically ameliorated in order to produce the stationary wide-band stochastic external excitation for the Monte Carlo simulating investigation of the nonlinear system. Both the theoretical solution and the numerical solution of the probabilistic properties of the system response are obtained, which are extremely coincident with each other. The numerical simulation and the theoretical computation all show that the time factor has a certain influence on the probability characteristics of the response. For example, the probabilistic distribution of the displacement tends to be scattered and the mean square displacement trends toward its steady-state value as time goes by. Of course the transient process to reach the steady-state value will obviously be shorter if the damping of the system is greater.

Transition probability density of a certain diffusion process concentrated on a finite spatial interval

1992 ◽  
Vol 29 (2) ◽  
pp. 334-342
Author(s):  
A. Milian
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
Transition Probability ◽  
Transition Probability Density ◽  
One Dimensional ◽  
Spatial Interval

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.

scholarly journals Numerical simulation of the two-locus Wright-Fisher stochastic differential equation with application to approximating transition probability densities

2020 ◽  
Author(s):  
Zhangyi He ◽  
Mark Beaumont ◽  
Feng Yu
Keyword(s):  
Differential Equation ◽  
Natural Selection ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Genetic Recombination ◽  
Transition Probability ◽  
Superior Performance ◽  
Transition Probability Density

AbstractOver the past decade there has been an increasing focus on the application of the Wright-Fisher diffusion to the inference of natural selection from genetic time series. A key ingredient for modelling the trajectory of gene frequencies through the Wright-Fisher diffusion is its transition probability density function. Recent advances in DNA sequencing techniques have made it possible to monitor genomes in great detail over time, which presents opportunities for investigating natural selection while accounting for genetic recombination and local linkage. However, most existing methods for computing the transition probability density function of the Wright-Fisher diffusion are only applicable to one-locus problems. To address two-locus problems, in this work we propose a novel numerical scheme for the Wright-Fisher stochastic differential equation of population dynamics under natural selection at two linked loci. Our key innovation is that we reformulate the stochastic differential equation in a closed form that is amenable to simulation, which enables us to avoid boundary issues and reduce computational costs. We also propose an adaptive importance sampling approach based on the proposal introduced by Fearnhead (2008) for computing the transition probability density of the Wright-Fisher diffusion between any two observed states. We show through extensive simulation studies that our approach can achieve comparable performance to the method of Fearnhead (2008) but can avoid manually tuning the parameter ρ to deliver superior performance for different observed states.

Sign in / Sign up

Export Citation Format

Share Document