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Non-autonomous fourth order random oscillating systems under the action of external periodical disturbances. (English) Zbl 1389.60070

Summary: The asymptotic behaviour of the non-autonomous oscillating system described by a differential equation of fourth order with small non-linear periodical external perturbations of “white noise”, non-centred and centred “Poisson noise” types is studied. Every term of the external perturbations has its own order of the small parameter \(\varepsilon\). If the small parameter is equal to zero, then general solution of the obtained non-stochastic fourth order differential equation has an oscillating part. We consider a given differential equation with external stochastic perturbations as the system of stochastic differential equations and study the limit behaviour of its solution at the time moment \(t/\varepsilon^k\), as \(\varepsilon\to 0\). A system of averaging stochastic differential equations is derived and its dependence on the order of the small parameter in every term of external perturbations is studied. The non-resonance and resonance cases, under the conditions that the characteristic equation has two non-equal real roots and two conjugate pure imaginary roots, are considered.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
60H40 White noise theory
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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