Construction of the stationary probability density for a family of SDOF strongly non-linear stochastic second-order dynamical systems

2008 ◽  
Vol 43 (7) ◽  
pp. 563-568 ◽  
Author(s):  
Z.L. Huang ◽  
X.L. Jin ◽  
J.Y. Li
2015 ◽  
Vol 3 (2) ◽  
pp. 176-183 ◽  
Author(s):  
Jiaorui Li ◽  
Shuang Li

AbstractSeveral observations in real economic systems have shown the evidence of non-Gaussianity behavior, and one of mathematical models to describe these behaviors is Poisson noise. In this paper, stationary probability density of a nonlinear business cycle model under Poisson white noise excitation has been studied analytically. By using the stochastic averaged method, the approximate stationary probability density of the averaged generalized FPK equations are obtained analytically. The results show that the economic system occurs jump and bifurcation when there is a Poisson impulse existing in the periodic economic system. Furthermore, the numerical solutions are presented to show the effectiveness of the obtained analytical solutions.


2010 ◽  
Vol 42 (04) ◽  
pp. 986-993 ◽  
Author(s):  
Muhamad Azfar Ramli ◽  
Gerard Leng

In this paper we generalize a bounded Markov process, described by Stoyanov and Pacheco-González for a class of transition probability functions. A recursive integral equation for the probability density of these bounded Markov processes is derived and the stationary probability density is obtained by solving an equivalent differential equation. Examples of stationary densities for different transition probability functions are given and an application for designing a robotic coverage algorithm with specific emphasis on particular regions is discussed.


2019 ◽  
Vol 26 (7-8) ◽  
pp. 532-539
Author(s):  
Lei Xia ◽  
Ronghua Huan ◽  
Weiqiu Zhu ◽  
Chenxuan Zhu

The operation of dynamic systems is often accompanied by abrupt and random changes in their configurations, which will dramatically change the stationary probability density function of their response. In this article, an effective procedure is proposed to reshape the stationary probability density function of nonlinear stochastic systems against abrupt changes. Based on the Markov jump theory, such a system is formulated as a continuous system with discrete Markov jump parameters. The limiting averaging principle is then applied to suppress the rapidly varying Markov jump process to generate a probability-weighted system. Then, the approximate expression of the stationary probability density function of the system is obtained, based on which the reshaping control law can be designed, which has two parts: (i) the first part (conservative part) is designed to make the reshaped system and the undisturbed system have the same Hamiltonian; (ii) the second (dissipative part) is designed so that the stationary probability density function of the reshaped system is the same as that of undisturbed system. The proposed law is exactly analytical and no online measurement is required. The application and effectiveness of the proposed procedure are demonstrated by using an example of three degrees-of-freedom nonlinear stochastic system subjected to abrupt changes.


2018 ◽  
Vol 32 (28) ◽  
pp. 1850313 ◽  
Author(s):  
Yong-Feng Guo ◽  
Fang Wei ◽  
Lin-Jie Wang ◽  
Jian-Guo Tan

In this paper, the Lévy noise-induced transition in an underdamped asymmetric bistable system is discussed. Lévy noise is generated  through the Janicki–Weron algorithm and the numerical solutions of system equation is obtained by the fourth-order Runge–Kutta method. Then the stationary probability density functions are obtained by solving the equation of system. The influence of the damped coefficient [Formula: see text], asymmetric parameter r of system, stability index [Formula: see text], skewness parameters [Formula: see text] and noise intensity D on the stationary probability density are analyzed. The numerical simulation results show that the asymmetric parameter r, stability index [Formula: see text], skewness parameters [Formula: see text] and noise intensity D can induce the phase transition. However, the phase transition cannot be induced by the damped coefficient [Formula: see text].


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