Moment Lyapunov exponents of a two-dimensional system under bounded noise parametric excitation

Journal of Sound and Vibration

Volumn 263, Issue 3, 2003, Pages 593-616

  Xie, W C ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFERENTIAL EQUATIONS; LYAPUNOV METHODS; PROBABILITY; RANDOM PROCESSES;

NOISE PARAMETRIC EXCITATION;

ACOUSTIC NOISE;

EID: 0038486195     PISSN: 0022460X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-460X(02)01068-4     Document Type: Article

References (20)

1. L. Arnold, E. Oeljeklaus, E. Pardoux, Almost sure and moment stability for linear Itô equations, in: L. Arnold, V. Wihstutz (Eds.), Lyapunov Exponents, Proceedings of a Workshop, Bremen, Germany, 1984, Lecture Notes in Mathematics, Vol. 1186, Springer, Berlin, 1986, pp. 85-125.
2. L. Arnold, W. Kliemann, E. Oeljeklaus, Lyapunov exponents of linear stochastic systems, in: L. Arnold, V. Wihstutz (Eds.), Lyapunov Exponents, Proceedings of a Workshop, Bremen, Germany, 1984, Lecture Notes in Mathematics, Vol. 1186, Springer, Berlin, 1986, pp. 129-159.
3. P.H. Baxendale, Moment stability and large deviations for linear stochastic differential equations, in: N. Ikeda (Ed.), Proceedings of the Taniguchi Symposium on Probabilistic Methods in Mathematical Physics, Kinokuniya, Tokyo, Japan, 1985, pp. 31-54.
4. L. Arnold, W. Kliemann, Large deviations of linear stochastic differential equations, in: H.J. Engelbert, W. Schmidt (Eds.), Stochastic Differential Systems, Lecture Notes in Control and Information Sciences, Vol. 96, Springer, Berlin, 1987, pp. 117-151.
5. P. Baxendale, D. Stroock, Large deviations and stochastic flows of diffeomorphisms, Probability Theory and Related Fields 80 (1988) 169-215.
6. L. Arnold, Random Dynamical Systems, Springer, Berlin, 1998.
7. L. Arnold, M.M. Doyle, N. Sri Namachchivaya, Small noise expansion of moment Lyapunov exponents for two-dimensional systems, Dynamics and Stability of Systems 12 (3) (1997) 187-211.
8. R.Z. Khasminskii, N. Moshchuk, Moment Lyapunov exponent and stability index for linear conservative system with small random perturbation, SIAM Journal of Applied Mathematics 58 (1) (1998) 245-256.
9. W.-C. Xie, Moment Lyapunov exponents of a two-dimensional system under real noise excitation, Journal of Sound and Vibration 239 (1) (2001) 139-155.
10. M.F. Dimentberg, Stability and subcritical dynamics of structures with spatially disordered travelling parametric excitation, Probabilistic Engineering Mechanics 7 (1992) 131-134.
11. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, 2nd Edition, Springer, Berlin, 1989.
12. R.L. Stratonovich, Topics in the Theory of Random Noise, Vol. II, Gordon and Breach, New York, 1967.
13. M.F. Dimentberg, Statistical Dynamics of Nonlinear and Time-Varying Systems, Wiley, New York, 1988.
14. W. Wedig, Analysis and simulation of nonlinear stochastic systems, in: W. Schiehlen (Ed.), Nonlinear Dynamics in Engineering Systems, Springer, Berlin, 1989, pp. 337-344.
15. Y.K. Lin, G.Q. Cai, Probabilistic Structural Dynamics: Advanced Theory and Applications, McGraw-Hill, New York, 1995.
16. S.T. Ariaratnam, Stochastic stability of viscoelastic systems under bounded noise excitation, in: A. Naess, S. Krenk (Eds.), IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, Kluwer Academic Publishers, Dordrecht, 1996, pp. 11-18.
17. R.Z. Khasminskii, Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems, Theory of Probability and Its Applications 12 (1967) 144-147 (English translation).
18. W. Wedig, Lyapunov exponent of stochastic systems and related bifurcation problems, in: S.T. Ariaratnam, G.I. Schuëller, I. Elishakoff (Eds.), Stochastic Structural Dynamics - Progress in Theory and Applications, Elsevier Applied Science, London, 1988, pp. 315-327.
19. E. Zauderer, Partial Differential Equations of Applied Mathematics, 2nd Edition, Wiley, New York, 1989.
20. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 5th Edition, Academic Press, Boston, 1994.