Asymptotic behavior of the stochastic Rayleigh-van der Pol equations with jumps. (English) Zbl 1322.60088

Huang, Zaitang; Chen, Chuntao

Abstr. Appl. Anal. 2013, Article ID 432704, 13 p. (2013).

Summary: We study the stability, attractors, and bifurcation of stochastic Rayleigh-van der Pol equations with jumps. We first established the stochastic stability and the large deviations results for the stochastic Rayleigh-van der Pol equations. We then examine the existence limit circle and obtain some new random attractors. We further establish stochastic bifurcation of random attractors. Interestingly, this shows the effect of the Poisson noise which can stabilize or unstabilize the system, which is significantly different from the classical Brownian motion process.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60F10	Large deviations
60J65	Brownian motion

Keywords:

stochastic Rayleigh-van der Pol equations; Poisson noise; stability; attractors; bifurcation; large deviations; Brownian motion

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