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Research Papers

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

[+] Author and Article Information
Mao Lin Deng, Yue Fu

Department of Mechanics,
Zhejiang University,
Hangzhou 310027, China

Zhi Long Huang

Department of Mechanics,
Zhejiang University,
Hangzhou 310027, China
e-mail: zlhuang@zju.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 17, 2014; final manuscript received May 15, 2014; accepted manuscript posted May 19, 2014; published online June 10, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(8), 081012 (Jun 10, 2014) (9 pages) Paper No: JAM-14-1117; doi: 10.1115/1.4027706 History: Received March 17, 2014; Revised May 15, 2014; Accepted May 19, 2014

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.

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Figures

Grahic Jump Location
Fig. 1

A particle moving in the bistable potential governed by Eqs. (35) and (36) with a = 1, b = 0.5

Grahic Jump Location
Fig. 2

The first-passage rate of a particle moving in the bistable potential governed by system Eq. (36). The symbol (○) denotes the result from digital simulation of the original system (36), the dashed line (- - - - -) from Kramers formula (40), and the solid line (——) from the asymptotic analytical solution (41); a = 1, b = a2/4ΔU, γ = 0.01, D = γkBT, kBT = 1.

Grahic Jump Location
Fig. 3

J(H,t|H0)/σ2(H) in Eq. (51) as a function of Hamiltonian H. Note that the function is independent of initial Hamiltonian H0 and varies slowly near the passage threshold Hc. The system parameters are ω1 = 1.414, ω2 = 1, γ = 0.1, λ = 1, and β = 0.05.

Grahic Jump Location
Fig. 4

u(H) in Eq. (20) as a function of Hamiltonian H. u(Ha) is the minimum value in the whole domain [0,Hc] and u(Hc) is the maximum value in the local domain [Ha,Hc]. The system parameters are the same as those in Fig. 3.

Grahic Jump Location
Fig. 7

The first-passage rate k(Hc) as a function of nonlinear stiffness parameter λ for system (44). The symbol (○) denotes the result from digital simulation of the original system (44), the dashed line (- - - - -) from numerically solving Pontryagin Eq. (14) with H0 = 0, and the solid line (——) from the asymptotic analytical solution (49). The system parameters are the same as those in Fig. 5 except that Hc = 1.2.

Grahic Jump Location
Fig. 8

The first-passage rate k(Hc) as a function of nonlinear damping parameter β for system (44). The symbol (○) denotes the result from digital simulation of the original system (44), the dashed line (- - - - -) from numerically solving Pontryagin Eq. (14) with H0 = 0, and the solid line (——) from the asymptotic analytical solution (49). The system parameters are the same as those in Fig. 7 except that λ = 0.

Grahic Jump Location
Fig. 9

The reliability function R(t|H0),R(t) for system (44). The symbol (○) denotes the result from digital simulation of the original system (44), the dashed line (- - - - -) from numerically solving backward Kolmogorov Eq. (9) with H0 = 0, and the solid line (——) from the asymptotic analytical solution (50). The system parameters are the same as those in Fig. 5 except that Hc = 1.

Grahic Jump Location
Fig. 10

The probability density of first-passage time p(T|H0),p(T) for system (44). The symbol (○) denotes the result from digital simulation of the original system (44), the dashed line (- - - - -) from numerically solving backward Kolmogorov Eqs. (9) and (12) with H0 = 0, and the solid line (——) from the asymptotic analytical solution (50). The system parameters are the same as those in Fig. 9.

Grahic Jump Location
Fig. 5

The first-passage rate k(Hc) for system (44) as a function of the nondimensional initial Hamiltonian H0/Hc for different passage threshold Hc. The four types of lines denote the results from numerically solving Pontryagin Eq. (14). The four types of symbols denote the results from digital simulation of the original system (44). The system parameters are the same as those in Fig. 3 except that D1 = 0.01 and D2 = 0.008.

Grahic Jump Location
Fig. 6

The first-passage rate k(Hc) as a function of passage threshold Hc for system (44). The symbol (○) denotes the result from digital simulation of the original system (44), the dashed line (- - - - -) from numerically solving Pontryagin Eq. (14) with H0=0, and the solid line (——) from the asymptotic analytical solution (49). The system parameters are the same as those in Fig. 5.

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