A Monte Carlo simulation method for determining the pth moment Lyapunov exponents of stochastic systems, which governs the pth 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.
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