Stochastic P bifurcation in a tri-stable van der Pol oscillator with fractional derivative excited by combined Gaussian white noises

PDF(1257 KB)

Journal of Vibration and Shock ›› 2021, Vol. 40 ›› Issue (16) : 275-280.

LI Yajie1, WU Zhiqiang2,3, LAN Qixun1, HAO Ying4, ZHANG Xiangyun5

Author information +

History +

Abstract

The stochastic P bifurcation behavior of tri-stability in a generalized van der Pol system with fractional derivative under additive and multiplicative Gaussian white noise excitations was investigated.Firstly, based on the minimal mean square error principle, the fractional derivative was found to be equivalent to a linear combination of damping and restoring forces, and the original system was simplified into an equivalent integer order system.Secondly, the stationary probability density function (PDF) of system amplitude was obtained by stochastic averaging, and according to the singularity theory, the critical parameters for stochastic P bifurcation of the system were found.Finally, the nature of stationary PDF curves of the system amplitude were qualitatively analyzed by choosing corresponding parameters in each region divided by the transition set curves.The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

Key words

combined Gaussian white noises / fractional derivative / stochastic P bifurcation / transition set / Monte Carlo simulation

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations

LI Yajie, WU Zhiqiang, LAN Qixun, HAO Ying, ZHANG Xiangyun. Stochastic P bifurcation in a tri-stable van der Pol oscillator with fractional derivative excited by combined Gaussian white noises[J]. Journal of Vibration and Shock, 2021, 40(16): 275-280

References

［1］XU M Y, TAN W C.Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion［J］.Science in China Series A: Mathematics，2001, 44(11):1387-1399.
［2］SABATIER J,AGRAWAL O P,MACHADO J A.Advancesin fractional calculus［M］.Dordrecht: Springer, 2007.
［3］PODLUBNY I.Fractional order systems and PID controller［J］.IEEE Transactions on Automatic Control, 1999,44(1):208-214.
［4］MONJE C A, CHEN Y Q, VINAGRE B M, et al.Fractional-order systems and controls: fundamentals and applications［M］.London: Springer-Verlag, 2010.
［5］BAGLEY R L, TORVIK J.Fractional calculus: a different approach to the analysis of viscoelastically damped structures［J］.AIAA Journal, 2012, 21(5): 741-748.
［6］BAGLEY R L, TORVIK P J.Fractional calculus in the transient analysis of viscoelastically damped structures［J］.AIAA Journal, 2012, 23(6):918-925.
［7］MACHADO J A T.Fractional order modelling of fractional-order holds［J］.Nonlinear Dynamics,2012,70(1):789-796.
［8］MACHADO J T.Fractional calculus: application in modeling and control［M］.New York: Springer, 2013.
［9］MACHADO J A T, COSTA A C, QUELHAS M D.Fractional dynamics in DNA［J］.Communications in Nonlinear Science & Numerical Simulation, 2011,16(8):2963-2969.
［10］RONG H W, WANG X D, XU W, et al.On double-peak probability density functions of a Duffing oscillator under narrow-band random excitations［J］.Acta Physica Sinica, 2005, 54(6):2557-2561.
［11］RONG H W, WANG X D, XU W, et al.On double peak probability density functions of duffing oscillator to combined deterministic and random excitations［J］.Applied Mathematics and Mechanics, 2006, 27(11):1569-1576.
［12］GU R C.Stochastic bifurcations in Duffing-van der Pol oscillator with Lévy stable noise［J］.Acta Physica Sinica, 2011, 60(6):1466-1467.
［13］XU Y, GU R C, ZHANG H Q, et al.Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise［J］.Physical Review E:Statistical Nonlinear & Soft Matter Physics, 2011, 83(2):056215.
［14］ZAKHAROVA A, VADIVASOVA T, ANISHCHENKO V, et al.Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator［J］.Physical Review E:Statistical Nonlinear & Soft Matter Physics,2010,81(1):011106.
［15］WU Z Q, HAO Y.StochasticP-bifurcations in tri-stable van der Pol-Duffing oscillator with multiplicative colored noise［J］.Acta Physica Sinica, 2015, 64(6):060501.
［16］CHEN L C, ZHU W Q.Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations［J］.International Journal of Non-Linear Mechanics, 2011, 46(10):1324-1329.
［17］HUANG Z L, JIN X L.Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative［J］.Journal of Sound & Vibration, 2009, 319(3):1121-1135.
［18］YANG Y G, XU W, SUN Y H, et al.Stochastic bifurcations in the nonlinear vibroimpact system with fractional derivative under random excitation［J］.Communications in Nonlinear Science & Numerical Simulation, 2017, 42:62-72.
［19］LI W, ZHANG M T, ZHAO J F.Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise［J］.Chinese Physics B, 2017(9):62-69.
［20］CHEN L C, WANG W H, LI Z S, et al.Stationary response of Duffing oscillator with hardening stiffness and fractional derivative［J］.International Journal of Non-Linear Mechanics, 2013, 48:44-50.
［21］CHEN L C, LI Z S, ZHUANG Q Q, et al.First-passage failure of single-degree-of-freedom nonlinear oscillators with fractional derivative［J］.Journal of Vibration & Control, 2013, 19(14):2154-2163.
［22］SHEN Y J, YANG S P, XING H J, et al.Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives［J］.Communications in Nonlinear Science & Numerical Simulation, 2012, 17(7):3092-3100.
［23］YANG Y G, XU W, SUN Y H, et al.Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation［J］.Chinese Physics B, 2016, 25(2):13-21.
［24］CHEN L C, ZHU W Q.Stochastic response of fractional-order van der Pol oscillator［J］.Theoretical and Applied Mechanics Letters, 2014, 4(1):68-72.
［25］SPANOS P D, ZELDIN B A.Random vibration of systems with frequency-dependent parameters or fractional derivatives［J］.Journal ofEngineering Mechanics,1997,123(3):290-292.
［26］朱位秋.随机振动［M］.北京:科学出版社,1992.
［27］凌复华.突变理论及其应用［M］.上海:上海交通大学出版社，1987.