A Monte Carlo simulation method for determining the moment Lyapunov exponents of stochastic systems, which governs the moment stability, is developed. Numerical results of two-dimensional systems under bounded noise and real noise excitations are presented to illustrate the approach.
1.
Oseledec
, Y. I.
, 1968
, “A Multiplicative Ergodic Theorem. Lyapunov Characteristic Number for Dynamical Systems
,” Trans. Mosc. Math. Soc.
, 19
, pp. 197
–231
.2.
Xie
, W.-C.
, 1998
, “Buckling Mode Localization in Rib-Stiffened Plates with Randomly Misplaced Stiffeners
,” Comput. Struct.
, 67
, pp. 175
–189
.3.
Wolf
, A.
, Swift
, J.
, Swinney
, H.
, and Vastano
, A.
, 1985
, “Determining Lyapunov Exponents from a Time Series
,” Physica D
, 16
, pp. 285
–317
.4.
Arnold
, L.
, 1984
, “A Formula Connecting Sample and Moment Stability of Linear Stochastic Systems
,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
, 44
, pp. 793
–802
.5.
Arnold, L., Oeljeklaus, E., and Pardoux, E., 1986, “Almost Sure and Moment Stability for Linear Ito^ Equations,” Lyapunov Exponents (Lecture Notes in Mathematics, 1186), L. Arnold and V. Wihstutz, eds., Springer-Verlag, Berlin, pp. 85–125.
6.
Arnold, L., Kliemann, W., and Oeljeklaus, E., 1986, “Lyapunov Exponents of Linear Stochastic Systems,” Lyapunov Exponents (Lecture Notes in Mathematics, 1186), L. Arnold and V. Wihstutz, eds., Springer-Verlag, Berlin, pp. 129–159.
7.
Baxendale, P. H., 1985, “Moment Stability and Large Deviations for Linear Stochastic Differential Equations,” Proceedings of the Taniguchi Symposium on Probabilistic Methods in Mathematical Physics, N. Ikeda, Kinokuniya, ed., Tokyo, Japan, 1985, pp. 31–54.
8.
Arnold, L., and Kliemann, W., 1987, “Large Deviations of Linear Stochastic Differential Equations,” Stochastic Differential Systems (Lecture Notes in Control and Information Sciences, 96), H. J. Englebert and W. Schmidt, eds., Springer-Verlag, Berlin, pp. 117–151.
9.
Baxendale
, P.
, and Stroock
, D.
, 1988
, “Large Deviations and Stochastic Flows of Diffeomorphisms
,” Probability Theory and Related Fields
, 80
, pp. 169
–215
.10.
Arnold, L., 1998, Random Dynamical Systems, Springer-Verlag, Berlin.
11.
Arnold
, L.
, Doyle
, M. M.
, and Sri Namachchivaya
, N.
, 1997
, “Small Noise Expansion of Moment Lyapunov Exponents for Two-Dimensional Systems
,” Dyn. Stab. Syst.
, 12
, No. 3
, pp. 187
–211
.12.
Khasminskii
, R. Z.
, and Moshchuk
, N.
, 1998
, “Moment Lyapunov Exponent and Stability Index for Linear Conservative System with Small Random Perturbation
,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
, 58
, No. 1
, pp. 245
–256
.13.
Sri Namachchivaya
, N.
, and Vedula
, L.
, 2000
, “Stabilization of Linear Systems by Noise: Application to Flow Induced Oscillations
,” Dyn. Stab. Syst.
, 15
(2
), pp. 185
–208
.14.
Sri Namachchivaya
, N.
, and Van Roessel
, H. J.
, 2001
, “Moment Lyapunov Exponent and Stochastic Stability of Two Coupled Oscillators Driven by Real Noise
,” ASME J. Appl. Mech.
, 68
, No. 6
, pp. 903
–914
.15.
Xie
, W.-C.
, 2001
, “Moment Lyapunov Exponents of a Two-Dimensional System under Real Noise Excitation
,” J. Sound Vib.
, 239
, No. 1
, pp. 139
–155
.16.
Xie
, W.-C.
, 2003
, “Moment Lyapunov Exponents of a Two-Dimensional System under Bounded Noise Parametric Excitation
,” J. Sound Vib.
, 263
, No. 3
, pp. 593
–616
.17.
Xie
, W.-C.
, 2001
, “Lyapunov Exponents and Moment Lyapunov Exponents of a Two-Dimensional Near-Nilpotent System
,” ASME J. Appl. Mech.
, 68
, No. 3
, pp. 453
–461
.18.
Xie
, W.-C.
, and So
, R. M. C.
, 2003
, “Numerical Determination of Moment Lyapunov Exponents of Two-Dimensional Systems,” ASME J. Appl. Mech., accepted for publication.19.
Kloeden, P. E., and Platen, E., 1992, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin.
Copyright © 2005
by ASME
You do not currently have access to this content.