Applied Mathematics and Mechanics

, Volume 20, Issue 9, pp 967–978 | Cite as

On the maximal Lyapunov exponent for a real noise parametrically excited Co-dimension two bifurcation system (I)

  • Liu Xianbin
  • Chen Dapeng
  • Chen Qiu
Article

Abstract

For a real noise parametrically excited co-dimension two bifurcation system on three-dimensional central manifold, a model of enhanced generality is developed in the present paper by assuming the real noise to be an output of a linear filter system, namely, a zero-mean stationary Gaussian diffusion process that satisfies the detailed balance condition. On such basis, asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are established by use of Arnold asymptotic analysis approach in parallel with the eigenvalue spectrum of Fokker-Planck operator.

Key words

real noise parametric excitation co-dimension two bifurcation detailed balance condition FPK equation singular boundary maximal Lyapunov exponent solvability condition 

CLC number

O211.63 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Arnold L., Wihstutz V.Lyapunov Exponents [M].Lecture Notes in Mathematics,1186. Berlin: Springer-Verlag, 1986.MATHGoogle Scholar
  2. [2]
    Arnold L., Crauel H, Eckmann J P.Lyapunov Exponents [M],Lecture Notes in Mathematics,1486. Berlin: Springer-Verlag, 1991.MATHGoogle Scholar
  3. [3]
    Liu Xianbin. Bifurcation behavior of stochastic mechanics system and its variational method [D]. Ph. D. Thesis. Chengdu:Southwest Jiaotong University, 1995. (in Chinese).Google Scholar
  4. [4]
    Liu Xianbin. Chen Qiu, Chen Dapeng. The researches on the stability and bifurcation of nonlinear stochastic dynamical systems [J].Advances in Mechanics, 1996,26 (4): 437–453 (in Chinese)MATHGoogle Scholar
  5. [5]
    Liu Xianbin, Chen Qiu. Advances in the researches on stochastic stability and stochastic bifurcation [R]. Invited paper of MMM-VII, Shanghai 1997. (in Chinese)Google Scholar
  6. [6]
    Khasminskii R Z,Stochastic Stability of Differential Equations [M], Alphen aan den Rijin, the Netherlands: Sijthoff and Noordhoff, 1980.MATHGoogle Scholar
  7. [7]
    Arnold L, Papanicolaou G, Wihstutz V. Asymptotic analysis of the Lyapunov exponents and rotation numbers of the random oscillator and applications [J].SIAM J Appl Math, 1986,46(3): 427–450.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Arnold L.. Lyapunov exponents of nonlinear stochastic systems [A]. In:, F Ziegler, G I Schueller eds.Nonlinear Stochastic Dynamic Engrg Systems, Berlin, New York: Springer-Verlag, 1987, 181–203.Google Scholar
  9. [9]
    Arnold L, Boxler P. Eigenvalues, bifurcation and center manifolds in the presence of noise [A]. In: C M Dafermos, G Ladas, G Papannicolaou eds,Differential Equations [M]. New York: Marcel Dekker Inc, 1990, 33–50.Google Scholar
  10. [10]
    Ariaratnam S T, Xie W C. Sensitivity of pitchfork bifurcation to stochastic perturbation [J].Dyna & Stab Sys, 1992,7 (3): 139–150.MATHMathSciNetGoogle Scholar
  11. [11]
    Ariaratnam S T, Xie W C. Lyapnov exponents and stochastic stability of coupled linear systems under real noise excitation [J].ASME J Appl Mech, 1992,59 (3): 664–673.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    Ariaratnam S T, Xie W C. Lyapunov exponents and stochastic stability of two-dimensional parametrically excited random systems [J].ASME J Appl Mech, 1993,60(5): 677–682.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    Kozin F.Stability of the Linear Stochaxtic Systems [M].Lecture Notes in Math,294. New York: Springer-Verlag, 1972, 186–229.Google Scholar
  14. [14]
    Namachchivaya Sri N, Ariaratnam S T. Stochastically perturbed Hoph bifurcation [J].Int J Nonlinear Mech, 1987,22 (5): 363–373.MATHCrossRefGoogle Scholar
  15. [15]
    Namachchivaya Sri N. Stochastic stability of a gyropendulum under random vertical support excitation [J].J Sound & Vib, 1987,119(2): 363–373.CrossRefGoogle Scholar
  16. [16]
    Namachchivaya Sri N. Hopf bifurcation in the presence of both parametric and external stochastic excitation [J].ASME J Appl Mech, 1998,55(4): 923–930.CrossRefGoogle Scholar
  17. [17]
    Namachchivaya Sri N, Talwar S. Maximal Lyapunov exponent and rotation number for stochastically peturbed co-dimension two bifurcation [J].J Sound & Vib, 1993,169 (3): 349–372.CrossRefGoogle Scholar
  18. [18]
    Liu Xianbin, Chen Qiu, Chen Dapeng. On the two bifurcations of a white-noise excited Hopf bifurcation system [J].Applied Mathematics and Mechanics (English Ed), 1997,18(9): 835–846.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Liu Xianbin, Chen Qiu, On the Hopf bifurcation system in the presence of parametric real noises [J].Acta Mechanica Sinica, 1997,29 (2): 158–166 (in Chinese)MathSciNetGoogle Scholar
  20. [20]
    Liu Xianbin, Chen Qiu, Sun Xunfang. On co-dimension 2 bifurcation system excited parametrically by white noise [J].Acta Mechanica Sinia, 1997,29(5) 563–572. (in Chinese)Google Scholar
  21. [21]
    Pardoux E, Wihstutz V. Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion [J].SIAM J Appl Math, 1988,48 (2): 442–457.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    Zhu Weiqiu.Stochastic Vibration [M]. Beijing: Science Press, 1992. (in Chinese)Google Scholar
  23. [23]
    Roy R V. Stochastic averaging of oscillators excited by coloured Gaussian processes [J].Int J Nonlinear Mech, 1994,29 (4): 461–475.CrossRefGoogle Scholar
  24. [24]
    Dygas M M K, Matkowsky B J, Schuss Z. Stochastic stability of nonlinear oscillators [J].SIAM J Appl Math, 1988,48(5):1115–1127.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 1999

Authors and Affiliations

  • Liu Xianbin
    • 1
  • Chen Dapeng
    • 2
  • Chen Qiu
  1. 1.Department of Applied Mechanics and EngineeringSouthwest Jiaotong UniversityChengduP R China
  2. 2.Institute of Engineering ScienceSouthwest Jiaotong UniversityChengduP R China

Personalised recommendations