Average first-passage time of a quasi-Hamiltonian Mathieu oscillator with parametric and forcing excitations

Abstract

a b s t r a c tA linear oscillator simultaneously subjected to stochastic forcing and parametric excitation is considered. The time required for this system to evolve from a low initial energy level until a higher energy state for the first time is a random variable. Its expectation satisfies the Pontryagin equation of the problem, which is solved with the asymptotic expansion method developed by Khasminskii. This allowed deriving closed-form expressions for the expected first passage time. A comprehensive parameter analysis of these solutions is performed. Beside identifying the important dimensionless groups governing the problem, it also highlights three important regimes which are called incubation, multiplicative and additive because of their specific features. Those three regimes are discussed with the parameters of the problem.

Second-order moment of the first passage time of a quasi-Hamiltonian oscillator with stochastic parametric and forcing excitations

“…3 (b). This formulation shows that the second-order moment of the first passage time is expressed as the product of also appears as a multiplicative factor, as expected from (13). Three regimes were identified through the average first passage time.…”

“…For this reason and those that have been mentioned in Ref. [13], we investigate the time required for the system to reach a certain displacement or amplitude, given an initial condition, or to reach a given energy barrier departing from a lower initial total internal energy level. This is known as a first passage problem.…”

“…Its mean value has already been investigated in Ref. [13]. The objective of this study is to determine the second-order statistical moment, in order to provide some information about the spread of this statistical distribution.…”

“…the system is quasi-Hamiltonian. The stochastic averaging of equation (4) over one revolution T = 2 using Itô differential rules and Wong-Zakaï correction terms [12] provides the averaged Itô equation governing the time-evolution of the Hamiltonian [13,34].…”

“…Accounting for the boundary conditions (11) and (12), it might be shown that the general solution of (10) for n = 2, with U 1 given by (13), is…”