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

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.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F10 Large deviations
60J65 Brownian motion
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[1] Applebaum, D., Lévy Processes and Stochastic Calculus. Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116 (2009), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1200.60001
[2] Bass, R. F., Stochastic differential equations with jumps, Probability Surveys, 1, 1-19 (2004) · Zbl 1189.60114
[3] Li, C. W.; Dong, Z.; Situ, R., Almost sure stability of linear stochastic differential equations with jumps, Probability Theory and Related Fields, 123, 1, 121-155 (2002) · Zbl 1019.34055
[4] Situ, R., Theory of Stochastic Differential Equations with Jumps and Applications. Theory of Stochastic Differential Equations with Jumps and Applications, Mathematical and Analytical Techniques with Applications to Engineering (2005), Berlin, Germany: Springer, Berlin, Germany · Zbl 1070.60002
[5] Figueroa-López, J. E.; Gong, R.; Houdré, C., Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy jumps, Stochastic Processes and their Applications, 122, 4, 1808-1839 (2012) · Zbl 1244.91089
[6] Sun, X.; Duan, J., Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Lévy processes, Journal of Mathematical Physics, 53, 7 (2012) · Zbl 1278.82045
[7] Arnold, L., Random Dynamical Systems. Random Dynamical Systems, Springer Monographs in Mathematics (1998), New York, NY, USA: Springer, New York, NY, USA · Zbl 0906.34001
[8] Huang, Z.; Yang, Q.; Cao, J., Stochastic stability and bifurcation for the chronic state in Marchuk’s model with noise, Applied Mathematical Modelling, 35, 12, 5842-5855 (2011) · Zbl 1228.93086
[9] Schenk-Hoppé, K. R., Random attractors—general properties, existence and applications to stochastic bifurcation theory, Discrete and Continuous Dynamical Systems, 4, 1, 99-130 (1998) · Zbl 0954.37026
[10] Li, J.; Xu, W., Stochastic stabilization of first-passage failure of Rayleigh oscillator under Gaussian white-noise parametric excitations, Chaos, Solitons and Fractals, 26, 5, 1515-1521 (2005) · Zbl 1076.60056
[11] Lin, Z.; Liu, J.; Zhang, W.; Niu, Y., Stabilization of interconnected nonlinear stochastic Markovian jump systems via dissipativity approach, Automatica, 47, 12, 2796-2800 (2011) · Zbl 1235.93251
[12] Xie, Y., The random attractors of stochastic Duffing-van der Pol equations with jumps, Chinese Journal of Applied Probability and Statistics, 26, 1, 9-23 (2010) · Zbl 1224.60152
[13] Touboul, J.; Wainribc, G., Bifurcations of stochastic differential equations with singular diffusion coefficients
[14] Fink, H.; Klüppelberg, C., Fractional Lévy-driven Ornstein-Uhlenbeck processes and stochastic differential equations, Bernoulli, 17, 1, 484-506 (2011) · Zbl 1284.60080
[15] Li, C. W.; Blankenship, G. L., Almost sure stability of linear stochastic systems with Poisson process coefficients, SIAM Journal on Applied Mathematics, 46, 5, 875-911 (1986) · Zbl 0613.60053
[16] Rong, S., On solutions of backward stochastic differential equations with jumps and applications, Stochastic Processes and their Applications, 66, 2, 209-236 (1997) · Zbl 0890.60049
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