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-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 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.

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.

Dynamics Response of a Hysteretic Structure Under Random Excitations

1993 ◽  
Author(s):  
Ismail I. Orabi
Keyword(s):  
White Noise ◽  
Transition Probability ◽  
Digital Simulation ◽  
Gaussian White Noise ◽  
Vertical Acceleration ◽  
Transition Probability Density ◽  
Vertical Excitation ◽  
Nonstationary Response ◽  
Dynamics Response ◽  
Range Of Values

Abstract The response of a hysteretic structure under horizontal and vertical random excitations is considered. The excitations are modeled by segments of stationary and nonstationary Gaussian white noise and filtered white noise processes. The linearization technique is used and the moments equations of the responses are evaluated. The transition probability density of the response is described and the associated second moment equations are derived. The transient and nonstationary response statistics for a range of values of parameters are obtained. A Monte-Carlo digital simulation study is performed. The results are compared with the theoretical findings and good agreements is observed. Particular attention is given to the amplification effects of the vertical acceleration. It is shown that the effect of the vertical excitation is usually insignificant, unless the load coefficient is quite large.

scholarly journals Stochastic Analysis of Predator–Prey Models under Combined Gaussian and Poisson White Noise via Stochastic Averaging Method

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1208
Author(s):  
Wantao Jia ◽  
Yong Xu ◽  
Dongxi Li ◽  
Rongchun Hu
Keyword(s):  
White Noise ◽  
Averaging Method ◽  
Gaussian Approximation ◽  
Perturbation Technique ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Direct Monte Carlo Simulation ◽  
Predator Population ◽  
Deterministic Models ◽  
Poisson White Noise

In the present paper, the statistical responses of two-special prey–predator type ecosystem models excited by combined Gaussian and Poisson white noise are investigated by generalizing the stochastic averaging method. First, we unify the deterministic models for the two cases where preys are abundant and the predator population is large, respectively. Then, under some natural assumptions of small perturbations and system parameters, the stochastic models are introduced. The stochastic averaging method is generalized to compute the statistical responses described by stationary probability density functions (PDFs) and moments for population densities in the ecosystems using a perturbation technique. Based on these statistical responses, the effects of ecosystem parameters and the noise parameters on the stationary PDFs and moments are discussed. Additionally, we also calculate the Gaussian approximate solution to illustrate the effectiveness of the perturbation results. The results show that the larger the mean arrival rate, the smaller the difference between the perturbation solution and Gaussian approximation solution. In addition, direct Monte Carlo simulation is performed to validate the above results.

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.

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