Asymptotic Analytical Solutions of First-Passage Rate to Quasi-Nonintegrable Hamiltonian Systems

2014 ◽  
Vol 81 (8) ◽  
Author(s):  
Mao Lin Deng ◽  
Yue Fu ◽  
Zhi Long Huang
Keyword(s):  
Analytical Solution ◽  
Hamiltonian Systems ◽  
First Passage Time ◽  
Reliability Function ◽  
Passage Time ◽  
Passage Rate ◽  
First Passage ◽  
High Passage ◽  
Associated Functions ◽  
Analytical Result

The first-passage problem of quasi-nonintegrable Hamiltonian systems subject to light linear/nonlinear dampings and weak external/parametric random excitations is investigated here. The motivation is to acquire asymptotic analytical solution of the first-passage rate or the mean first-passage time based on the averaged Itô stochastic differential equation for quasi-nonintegrable Hamiltonian systems. By using the probability current equation and the Laplace integral method, a new method is proposed to obtain the asymptotic analytical expressions for the first-passage rate in the case of high passage threshold. The associated functions such as the reliability function and the probability density function of first-passage time can then be obtained from the first-passage rate. High passage threshold is the crucial condition for the validity of the proposed method. The random bistable oscillator is studied as an illustrative example using the method. The analytical result obtained from the asymptotic analysis shows its consistency with the Kramers formula. A coupled two-degree-of-freedom (2DOF) nonlinear oscillator subjected to stochastic excitations is studied to illustrate the procedure of acquiring the asymptotic analytical solution. The results obtained from the analytical solution agree well with those from numerical simulation, which verifies the accuracy of the proposed method.

scholarly journals First-Passage Failure of Quasi-Integrable Hamiltonian Systems

2002 ◽  
Vol 69 (3) ◽  
pp. 274-282 ◽  
Author(s):  
W. Q. Zhu ◽  
M. L. Deng ◽  
Z. L. Huang
Keyword(s):  
Hamiltonian Systems ◽  
First Passage Time ◽  
Digital Simulation ◽  
Reliability Function ◽  
Passage Time ◽  
Kolmogorov Equation ◽  
Conditional Probability Density ◽  
Integrable Hamiltonian Systems ◽  
First Passage ◽  
First Passage Failure

The first-passage failure of quasi-integrable Hamiltonian systems (multidegree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is investigated. The motion equations of such a system are first reduced to a set of averaged Ito^ stochastic differential equations by using the stochastic averaging method for quasi-integrable Hamitonian systems. Then, a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, and the conditional probability density and moments of first-passage time are obtained by solving these equations with suitable initial and boundary conditions. Two examples are given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the procedure.

Reliability of Duffing-van der Pol Oscillator with Delayed Feedback Control under Wide-Band Noise Excitation

2013 ◽  
Vol 302 ◽  
pp. 717-722 ◽  
Author(s):  
Ting Zhang ◽  
Chang Shui Feng ◽  
Qiao Yi Wang
Keyword(s):  
Feedback Control ◽  
First Passage Time ◽  
Wide Band ◽  
Reliability Function ◽  
Passage Time ◽  
Delayed Feedback ◽  
Van Der Pol Oscillator ◽  
Delayed Feedback Control ◽  
Control Force ◽  
First Passage

The first passage type reliability of Duffing-van der Pol oscillator with time-delayed feedback control under wide-band noise excitations is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method, from which a backward Kolmogorov equation governing the conditional reliability function and a Pontryagin equation governing the conditional mean of the first passage time are established. Finally, the conditional reliability function and the conditional mean of first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. The effect of time delay in feedback control force on the reliability is analyzed. The theoretical results are well verified through digital simulation.

First passage time for the state of exact chemical equilibrium

1980 ◽  
Vol 45 (3) ◽  
pp. 777-782 ◽  
Author(s):  
Milan Šolc
Keyword(s):  
Chemical Equilibrium ◽  
First Passage Time ◽  
Passage Time ◽  
The State ◽  
Recurrence Time ◽  
First Passage ◽  
First Order ◽  
The Mean ◽  
Passage Times ◽  
Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

scholarly journals Credit Gap Risk in a First Passage Time Model with Jumps

2009 ◽  
Author(s):  
Natalie Packham ◽  
Lutz Schloegl ◽  
Wolfgang M. Schmidt
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Time Model ◽  
Gap Risk ◽  
Credit Gap

Spectral Theory for Skip-Free Markov Chains

1989 ◽  
Vol 3 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Markov Chains ◽  
Spectral Theory ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage Times ◽  
Simple Relationship ◽  
First Passage ◽  
Birth And Death Processes ◽  
The Laplace Transform ◽  
Passage Times

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

Sign in / Sign up

Export Citation Format

Share Document