Abstract
In this paper, the stochastic stability and bifurcation of a stochastic differential equation modeling a hexagonal governor system are investigated. More precisely, we introduced the stochasticity into the model based on the parameter perturbation, and simplified the stochastic hexagonal governor system by using the stochastic center manifold theory and stochastic average theory. Besides, we investigated the local stochastic stability and global stochastic stability of the stochastic hexagonal governor system through the use of the Lyapunov exponent and singular boundary theory. And based on the invariant measure and stationary probability density, we studied the stochastic bifurcation of the stochastic hexagonal governor system. Finally, we obtained some new criteria to ensure the stochastic pitchfork bifurcation and P-bifurcation of the stochastic hexagonal governor system.
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Seydel, R.: Practical Bifurcation and sTability Analysis: from Equilibrium to Chaos, pp. 68–93. Springer, New York (1994)
Chen, L.Q., Liu, Y.Z.: A modified exact linearization control for chaotic oscillators. Nonlinear Dyn. 20(4), 309–317 (1999)
WangCai, Ding: Hopf bifurcation of flywheel governor with feedback control device. J. Southwest Jiaotong Univ. 36(6), 624–628 (2001)
Ge, Z., Lee, C.I.: Ci. L. Non-linear dynamics and control of chaos for a rotational machine with a hexagonal centrifugal governor with a spring. J. Sound Vib. 262(4), 845–864 (2003)
Shufen, Zhang, Liuding, Tang: Routh-Hurwitz method of stability analysis for rotat ional speed modulator of centrifugal. J. Henan Univ. Sci. Technol. 26(5), 29–31 (2005)
Chen, H.K., Ge, Z.M.: Bifurcations and chaos of a two-degree-of-freedom dissipative gyroscope. Chaos Solitons Fractals 24(1), 125–136 (2005)
Sotomayor, J., Mello, L.F., Braga, D.C.: Stability and hopf bifurcation in the watt governor system. Commun. Appl. Nonlinear Anal. 13(4), 1–17 (2006)
Beltrami, e: Mathematics for Dynamic Modeling. Academic Press Inc, New York (1987)
Wang, J.Y.: Chaos and its speed feedback control in centrifugal flywheel governor. J. Mech. Strength 32(3), 509–512 (2010)
Wang, J., Wang, H., Guo, L.: Anti-control of chaos in mechanical centrifugal governor system. J. Inf. Comput. Sci. 9(17), 5127–5133 (2012)
Peng, J.K., Yu, J.N., Zhang, L., et al.: Study on synchronization of the centrifugal flywheel governor system. Appl. Mech. Mater. 433–435, 21–29 (2013)
Wen, G.L., Xu, H.D., Lv, Z.Y., et al.: Anti-controlling Hopf bifurcation in a type of centrifugal governor system. Nonlinear Dyn. 81(1–2), 1–12 (2015)
Luo, S., Hou, Z., Zhang, T.: Performance enhanced design of chaos controller for the mechanical centrifugal flywheel governor system via adaptive dynamic surface control. Aip Adv. 6(9), 1881–1888 (2016)
Svishchuk, A.V., Svishchuk, M.Y.: Stochastic stability of processes determined by poisson differential equations with delay. Ukr. Math. J. 54(3), 511–518 (2002)
Zhu, W.Q.: Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems. Int. J. NonLinear Mech. 39(4), 569–579 (2004)
Larsen, J.W., Iwankiewicz, R., Nielsen, S.R.K.: Nonlinear stochastic stability analysis of wind turbine wings by Monte Carlo simulations. Probab. Eng. Mech. 22(2), 181–193 (2007)
Li, Y., Zhang, W., Liu, X.: Stability of Nonlinear Stochastic Discrete-Time Systems. J. Appl. Math. 2013(4), 993–1000 (2013)
Zhao, J.Q., Liu, M.X., Yang-Jun, M.A., et al.: Stochastic stability and bifurcation of an SI epidemic model with double noises. Appl. Math. Mech. 34(12), 1300–1310 (2013)
Zhang, B., Jing, Z., Liu, W.W.: Stochastic stability of a suspended wheelset system under gauss white noise. J. Vib. Shock 34(19), 49–56 (2015)
Kim, S.H., Nguyen, N.H.A.: Stochastic stability analysis of semi-Markovian jump linear systems via a relaxation technique for time-varying transition rates, pp. 995–998 (2015)
Zhu, W.: Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation. Science Press, Beijing (2003)
Xu, Y., Feng, J., Li, J.J., et al.: Stochastic bifurcation for a tumor-immune system with symmetric Lévy noise. Physica A 392(392), 4739–4748 (2013)
Hao, Y., Wu, Z.: Stochastic P-bifurcation of tri-stable Van der Pol-Duffing oscillator. Chin. J. Theor. Appl. Mech. 45(2), 257–264 (2013)
Ma, S.J., Dong, D., Yang, M.S.: Stochastic Hopf bifurcation analysis in a stochastic lagged logistic discrete-time system with Poisson distribution coefficient. Nonlinear Dyn. 80(1–2), 269–279 (2014)
Zhu, Z., Zhang, W., Xu, J.: Stochastic bifurcation characteristics of SMA intravascular stent subjected to radial and axial excitations. BioMed. Mater. Eng. 24(6), 2465–2473 (2014)
Yang, J.H., Sanjuán, M.A.F., Liu, H.G., et al.: Stochastic P-bifurcation and stochastic resonance in a noisy bistable fractional-order system. Commun. Nonlinear Sci. Numer. Simul. 41, 104–117 (2016)
Luo, C., Guo, S.: Stability and bifurcation of two-dimensional stochastic differential equations with multiplicative excitations. Bull. Malays. Math. Sci. Soc. 1–23 (2016)
Zhang, J.G., Mello, L.F., Chu, Y.D., et al.: Hopf bifurcation in an hexagonal governor system with a spring. Commun. Nonlinear Sci. Numer. Simul. 15(3), 778–786 (2010)
Yamapi, R., Filatrella, G.: Strange attractors and synchronization dynamics of coupled Van der Pol-Duffing oscillators. Commun. Nonlinear Sci. Numer. Simul. 13(6), 1121–1130 (2008)
Li, X.F., Chlouverakis, K.E., Xu, D.L.: Nonlinear dynamics and circuit realization of a new chaotic flow: a variant of Lorenz, Chen and Lü. Nonlinear Anal. 10(4), 2357–2368 (2009)
Wang, S., Chang, Y., Li, X., et al.: Parameter identification for a class of nonlinear chaotic and hyperchaotic flows. Nonlinear Anal. 11(1), 423–431 (2010)
Borisov, A.V., Kazakov, A.O., Kuznetsov, S.P.: Nonlinear dynamics of the rattleback: a nonholonomic model. Phys. Uspekhi 57(5), 453 (2014)
Savi, M.A.: Nonlinear dynamics and chaos in shape memory alloy systems. Int. J. NonLinear Mech. 70, 2–19 (2015)
Miandoab, E.M., Yousefi-Koma, A., Pishkenari, H.N., et al.: Study of nonlinear dynamics and chaos in MEMS/NEMS resonators. Commun. Nonlinear Sci. Numer. Simul. 22(1), 611–622 (2015)
Ling, G., Guan, Z.H., Liao, R.Q., et al.: Stability and bifurcation analysis of cyclic genetic regulatory networks with mixed time delays. SIAM J. Appl. Dyn. Syst. 14(1), 202–220 (2015)
Zhang, Y., Zheng, Y., Zhao, F., et al.: Dynamical analysis in a stochastic bioeconomic model with stage-structuring. Nonlinear Dyn. 84(2), 1113–1121 (2016)
Khas’ minskii, R.Z.: Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems. Theory Probab. Appl. 12(1), 144–147 (1967)
Lim, Y., Cai, G.: Probabilistic Structural Dynamics. Mcgraw-hill Professional Publishing, New York (2004)
Namachchivaya, N.: Stochastic bifurcation. Appl. Math. Comput. 38, 101–159 (1990)
Khasminskii, R.: On the principle of averaging for Ito stochastic differential equations. Kybernetika (Prague) 4, 260–279 (1968)
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The authors gratefully acknowledge support from the National Natural Science Foundation of China (No. 61364001) and Lanzhou Talent innovation and Entrepreneurship Project (No. 2015-RC-3).
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Zhang, J., Chu, Y., Du, W. et al. The invariant measure and stationary probability density computing model based analysis of the governor system. Cluster Comput 20, 1437–1447 (2017). https://doi.org/10.1007/s10586-017-0817-4
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DOI: https://doi.org/10.1007/s10586-017-0817-4