Abstract
The asymptotic behavior of a non-autonomous oscillating system described by a differential equation of the fourth order with small non-linear periodical external perturbations of “white noise”, non-centered and centered “Poisson noise” types is studied. Each term of external perturbations has own order of a small parameter e. If the small parameter is equal to zero, then the general solution of the obtained non-stochastic fourth order differential equation has an oscillating part. We consider the given differential equation with external stochastic perturbations as the system of stochastic differential equations and study the limit behavior of its solution at the time moment t/ek, as e → 0. The system of averaging stochastic differential equations is derived and its dependence on the order of the small parameter in each term of external perturbations is studied. The non-resonance and resonance cases are considered.
Highlights
Studying of oscillation processes has a great importance in different areas of mechanics, physics, technics, and economics
We investigate the asymptotic behavior of the oscillating system (1), as ε → 0, in the case when the characteristic equation has multiple real root and two conjugate pure imaginary roots (Theorem 3), and in the case of two pairs of imaginary adjoined roots of the characteristic equation (Theorem 4)
We can solve the system of linear equations (3) with respect to (N1(t), N2(t), A1(t), A2(t)), where Ni(t) = Ci(t)e−ηit, i = 1, 2 and using the Ito formula we derive the system of stochastic differential equations: 1 dN1(t) = −η1N1(t) dt + (η2 − η1)(η12 + ω2) dH(t), 1 dN2(t) = −η2N2(t) dt − (η2 − η1)(η22 + ω2) dH(t), dA1(t) η2) cos ωt + (ω2 − η1η2) sin ωt ω(η12 + ω2)(η22 + ω2)
Summary
Studying of oscillation processes has a great importance in different areas of mechanics, physics, technics, and economics. C(t) = (C1(t), C2(t), A1(t), A2(t)), Θ(t) = (e−η1t, e−η2t, cos ωt, sin ωt), and let us consider the following representation of the solution y(t) to the system (2): (
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