# Non-Autonomous Random Oscillating Systems of Fourth Order under Small Periodical External Perturbations with Jumps

### 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 ε. 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/εk, as ε → 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.### References

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*Statistics, Optimization & Information Computing*,

*7*(1), 133-149. https://doi.org/10.19139/soic.v7i1.459

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