Stochastic bifurcations of a fractional-order vibro-impact system driven by additive and multiplicative Gaussian white noises. (English) Zbl 1477.34085
It has been revealed that several natural systems are explained with differential fractional-order equations, where a fractional derivative order is utilized. As the stochastic disturbances are inevitable such as strong winds, earthquakes, and ocean waves, in recent years, stochastic differential fractional-order equations have become more and more important and stimulating to researchers due to their successful and possible applications in numerous arenas. Vibro-impact systems are used broadly in modelling machine dynamics, vibration engineering, and structural mechanics. Study of vibro-impact systems requires the investigation of mathematical models with discontinuities and shows their behaviour as intensely non-linear. The debate on the stochastic systems with both vibro-impact factors and fractional derivative element is uncommon. This paper is involved with the stochastic bifurcation of a fractional-order vibro-impact system powered by additive and multiplicative Gaussian white noises. There are two challenges to study the stationary response of the fractional vibro-impact systems under Gaussian white noises. The first one is how to deal with the discontinuity of the original system. The second one is how to get the explicit expression of the averaged drift coefficient when we utilize the stochastic averaging method. These two challenges have been solved in this paper by the non-smooth transformations and stochastic averaging method.
Reviewer: Seifedine Kadry (Beirut)
MSC:
| 34F10 | Bifurcation of solutions to ordinary differential equations involving randomness |
| 34A08 | Fractional ordinary differential equations |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34A36 | Discontinuous ordinary differential equations |
| 34F05 | Ordinary differential equations and systems with randomness |
| 34C23 | Bifurcation theory for ordinary differential equations |
Keywords:
fractional derivative; bifurcations; random process; stochastic fractional-order systems; vibro-impact factors; white noiseReferences:
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