Skip to main content
SpringerLink
Log in

The invariant measure and stationary probability density computing model based analysis of the governor system

  • Published:
Cluster Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Seydel, R.: Practical Bifurcation and sTability Analysis: from Equilibrium to Chaos, pp. 68–93. Springer, New York (1994)

    MATH  Google Scholar 

  2. Chen, L.Q., Liu, Y.Z.: A modified exact linearization control for chaotic oscillators. Nonlinear Dyn. 20(4), 309–317 (1999)

    Article  MATH  Google Scholar 

  3. WangCai, Ding: Hopf bifurcation of flywheel governor with feedback control device. J. Southwest Jiaotong Univ. 36(6), 624–628 (2001)

    Google Scholar 

  4. 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)

    Article  MATH  Google Scholar 

  5. 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)

    Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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)

    MathSciNet  MATH  Google Scholar 

  8. Beltrami, e: Mathematics for Dynamic Modeling. Academic Press Inc, New York (1987)

    MATH  Google Scholar 

  9. Wang, J.Y.: Chaos and its speed feedback control in centrifugal flywheel governor. J. Mech. Strength 32(3), 509–512 (2010)

    Google Scholar 

  10. Wang, J., Wang, H., Guo, L.: Anti-control of chaos in mechanical centrifugal governor system. J. Inf. Comput. Sci. 9(17), 5127–5133 (2012)

    Google Scholar 

  11. 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)

    Article  Google Scholar 

  12. 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)

    MATH  Google Scholar 

  13. 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)

    Article  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhu, W.Q.: Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems. Int. J. NonLinear Mech. 39(4), 569–579 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

    Article  Google Scholar 

  17. Li, Y., Zhang, W., Liu, X.: Stability of Nonlinear Stochastic Discrete-Time Systems. J. Appl. Math. 2013(4), 993–1000 (2013)

    MathSciNet  Google Scholar 

  18. 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)

    Google Scholar 

  19. 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)

    Google Scholar 

  20. 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)

  21. Zhu, W.: Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation. Science Press, Beijing (2003)

    Google Scholar 

  22. 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)

    Article  MathSciNet  Google Scholar 

  23. 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)

    Google Scholar 

  24. 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)

    MathSciNet  MATH  Google Scholar 

  25. 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)

    Google Scholar 

  26. 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)

    Article  MathSciNet  Google Scholar 

  27. Luo, C., Guo, S.: Stability and bifurcation of two-dimensional stochastic differential equations with multiplicative excitations. Bull. Malays. Math. Sci. Soc. 1–23 (2016)

  28. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Borisov, A.V., Kazakov, A.O., Kuznetsov, S.P.: Nonlinear dynamics of the rattleback: a nonholonomic model. Phys. Uspekhi 57(5), 453 (2014)

    Article  Google Scholar 

  33. Savi, M.A.: Nonlinear dynamics and chaos in shape memory alloy systems. Int. J. NonLinear Mech. 70, 2–19 (2015)

    Article  Google Scholar 

  34. 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)

    Article  MATH  Google Scholar 

  35. 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)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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)

  37. Khas’ minskii, R.Z.: Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems. Theory Probab. Appl. 12(1), 144–147 (1967)

    Article  MathSciNet  Google Scholar 

  38. Lim, Y., Cai, G.: Probabilistic Structural Dynamics. Mcgraw-hill Professional Publishing, New York (2004)

    Google Scholar 

  39. Namachchivaya, N.: Stochastic bifurcation. Appl. Math. Comput. 38, 101–159 (1990)

    MathSciNet  MATH  Google Scholar 

  40. Khasminskii, R.: On the principle of averaging for Ito stochastic differential equations. Kybernetika (Prague) 4, 260–279 (1968)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

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).

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, 730070, China

    Jiangang Zhang, Yandong Chu, Jiarong Lu & Hongwei Luo

  2. School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, 730070, China

    Wenju Du

  3. Department of Information Engineering, Gansu forestry technological College, Tianshui, Gansu, 741020, China

    Hongwei Luo

Authors
  1. Jiangang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yandong Chu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenju Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiarong Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Hongwei Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiangang Zhang.

Ethics declarations

Conflicts of interest

The authors have declared that no conflict of interest exists.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10586-017-0817-4

Keywords

Search

Navigation