Applied Mathematics and Mechanics

, Volume 20, Issue 10, pp 1067–1074 | Cite as

On the maximal lyapunov exponent for a real noise parametrically excited co-dimension two bifurcation system (II)

  • Liu Xianbin
  • Chen Qiu
  • Chen Dapeng


For a co-dimension two bifurcation system on a three-dimensional central manifold, which is parametrically excited by a real noise, a rather general model is obtained by assuming that the real noise is an output of a linear filter system-a zeromean stationary Gaussian diffusion process which satisfies detailed balance condition. By means of the asymptotic analysis approach given by L. Arnold and the expression of the eigenvalue spectrum of Fokker-Planck operator, the asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are obtained.

Key words

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

CLC number



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ito K, McKean H P, Jr.Diffusion Processes and Their Sample Paths[M]. New York: Springer-Verlag, 1965.Google Scholar
  2. [2]
    Karlin S, Taylor H M.A Second Course in Stochastic Processes [M]. New York: Academic Press, 1981.Google 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): 437–452. (in Chinese)zbMATHGoogle Scholar
  5. [5]
    Liu Xianbin, Chen Qiu, Sun Xunfang. On co-dimension 2 bifurcation system excited parametrically by white noise[J].Acta Mechanica Sinica, 1997,29(5): 563–572. (in Chinese)MathSciNetGoogle Scholar
  6. [6]
    Zhu Weiqiu.Stochastic Vibration[M]. Beijing: Science Press, 1992. (in Chinese)Google Scholar
  7. [7]
    Lin Y K, Cai G Q. Stochastic stability of nonlinear system[J].Int J Nonlinear Mech, 1994,29(4): 539–555.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    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.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    Arnold L, Wihstutz V.Lyapunov Exponents[M].Lecture Notes in Mathematics, 1186, Berlin, Springer-Verlag, 1986.Google Scholar
  10. [10]
    Arnold L, Papanicolaou G, Wihstutz V. Asympototic analysis of the Lyapunov exponents and rotation numbers of the random oscilltor and applications[J].SIAM J Appl Math, 1986,46(3): 427–450.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Liu Xianbin, Chen Dapang, Chen Qiu. On the maximal Lyapunov exponent for a real noise parametrically excited co-dimension two bifurcation system (I) [J].Applied Mathematics and Mechanics (English Ed), 1999,20(9): 967–978.MathSciNetzbMATHGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 1999

Authors and Affiliations

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

Personalised recommendations