Moment Liapunov Exponent of a Three-Dimensional System Under Bounded Noise Excitation

FANG Ci-jun; YANG Jian-hua; LIU Xian-bin

doi:10.3879/j.issn.1000-0887.2012.05.002

Volume 33 Issue 5

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2012 > 33(5): 526-538

FANG Ci-jun, YANG Jian-hua, LIU Xian-bin. Moment Liapunov Exponent of a Three-Dimensional System Under Bounded Noise Excitation[J]. Applied Mathematics and Mechanics, 2012, 33(5): 526-538. doi: 10.3879/j.issn.1000-0887.2012.05.002

Citation:

PDF( 1103 KB)

Moment Liapunov Exponent of a Three-Dimensional System Under Bounded Noise Excitation

doi: 10.3879/j.issn.1000-0887.2012.05.002

¹State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics and Astronautics,Nanjing 210016, P.R.China;²School of Science, Hubei University of Technology, Wuhan 430068, P.R.China;³School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou,Jiangsu 221116, P.R.China

Received Date: 2011-09-06
Rev Recd Date: 2012-02-16
Publish Date: 2012-05-15

Abstract

Abstract

The moment Liapunov exponent of a co-dimension two bifurcation system was evaluated, which was on a three-dimensional central manifold and was subjected to a parametric excitation by a bounded noise. Based on the theory of the stochastic dynamics, the eigenvalue problem governing the moment Liapunov exponent was established. Through a singular perturbation method, the explicit asymptotic expressions or numerical results of the second-order, weak noise expansions of the moment Liapunov are obtained for two cases. Then the effects of the bounded noise and the parameters of the system on the moment Liapunov exponent and the stability index were investigated. It is found that the stochastic stability of the system can be strengthened by the bounded noise.
- bounded noise,
- moment Liapunov exponent,
- stability index

FullText(HTML)

References(15)

References

[1]	Arnold L. A formula connecting sample and moment stability of a linear stochastic system[J]. SIAM Journal of Applied Mathematics, 1984, 44(4): 793-802.
[2]	Arnold L. Random Dynamical Systems[M]. Berlin: Springer, 1998.
[3]	Khasminkii R, Moshchuk N. Moment Lyapunov exponent and stability index for linear conservative system with small random perturbation[J]. SIAM Journal of Applied Mathematics, 1998, 58(1): 245-256.
[4]	Namachichivaya N S, van Roessel H J. Moment Lyapunov exponent and stochastic stability of two coupled oscillators driven by real noise[J]. ASME Journal of Applied Mechanics, 2001, 68(6): 903-914.
[5]	Liu X B, Liew K M. On the stability properties of a van der Pol-Duffing oscillator that is driven by a real noise[J]. Journal of Sound and Vibration, 2005, 285 (1/2):27-49.
[6]	Xie W C. Moment Liapunov exponents of a two-dimensional system under bounded noise parametric excitation[J]. Journal of Sound and Vibration, 2003, 263(3):593-616.
[7]	Xie W C, Ronalad M C So. Parametric resonance of a two-dimensional system under bounded noise excitation[J]. Nonlinear Dynamics, 2004, 36(2/4):437-453.
[8]	Xie W C. Moment Lyapunov exponents of a two-dimensional system under both harmonic and white noise parametric excitations[J]. Journal of Sound and Vibration, 2006, 289(1/2): 171-191.
[9]	Xie W C. Moment Lyapunov exponents of a two-dimensional system under combined harmonic and real noise excitations[J]. Journal of Sound and Vibration, 2007, 303(1/2): 109-134.
[10]	Zhu J Y, Xie W C. Parametric resonance of a two degrees-of-freedom system induced by bounded noise[J]. ASME Journal of Applied Mechanics, 2009, 76(4): 041007.
[11]	Guckenheimer G, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields[M]. New York: Springer-Verlag, 1983.
[12]	Stratonovich R L. Topics in the Theory of Random Noise[M]. Vol Ⅱ. New York: Gordon and Breach, 1967.
[13]	Lin Y K, Cai G Q. Probabilistic Structural Dynamics, Advanced Theory and Applications[M]. New York: McGraw-Hill, 1995.
[14]	Ariaratnam S T. Stochastic stability of viscoelastic systems under bounded noise excitation[C]Naess A, Krenk S.IUTAM Symposium on Advences in Nonlinear Stochastic Mechnics. The Netherlands: Kluwer, Dordrecht, 1996: 11-18.
[15]	Zauderer E.Partial Differential Equations of Applied Mathematics[M].2nd ed. New York: Wiley, 1989.