This paper examines the almost-sure asymptotic stability condition of a linear multiplicative stochastic system, which is a linear part of a co-dimension two-bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by an ergodic real noise. The excitation is assumed to be an integrable function of an n-dimensional Ornstein-Uhlenbeck vector process which is the output of a linear filter system, while both the detailed balance condition and the strong mixing condition are removed. Through a perturbation method and the spectrum representations of the Fokker Planck operator and its adjoint operator of the linear filter system, the explicit asymptotic expressions of the maximal Lyapunov exponent for three case studies, in which different forms of the coefficient matrix included in the noise excitation term are assumed, are obtained.

Issue Section:
Technical Papers
Keywords:
stochastic systems, asymptotic stability, Lyapunov matrix equations, stochastic processes, linear systems, bifurcation, Fokker-Planck equation, perturbation techniques, spectral analysis, Gaussian noise
Topics:
Bifurcation, Density, Diffusion (Physics), Diffusion processes, Eigenvalues, Emission spectroscopy, Filters, Noise (Sound), Probability, Spectroscopy, Stability, Stochastic systems, Excitation, Dimensions, Eigenfunctions
1.
Khasminskii
,
R. Z.
,
1967
, “
Necessary and Sufficient Conditions for the Asymptotic Stability of Linear Stochastic Systems
,”
Theor. Probab. Appl.
,
12
, pp.
144
–
147
.
2.
Mitchell
,
P. R.
, and
Kozin
,
F.
,
1974
, “
Sample Stability of Second Order Linear Differential Equations With Wide Band Noise Coefficients
,”
SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
,
27
, pp.
571
–
605
.
3.
Nishoka
,
K.
,
1976
, “
On the Stability of Two-Dimensional Linear Stochastic Systems
,”
Kodai Math. Sem. Rep.
,
27
, pp.
211
–
230
.
4.
Ariaratnam
,
S. T.
, and
Xie
,
W. C.
,
1992
, “
Lyapunov Exponents and Stochastic Stability of Coupled Linear Systems Under Real Noise Excitation
,”
ASME J. Appl. Mech.
,
59
, pp.
664
–
673
.
5.
Xie
,
W. C.
,
2002
, “
Moment Lyapunov Exponents of Two-Dimensional Viscoelastic System Under Bounded Noise Excitation
,”
ASME J. Appl. Mech.
,
69
, pp.
346
–
357
.
6.
Arnold
,
L.
,
Papanicolaou
,
G.
, and
Wihstutz
,
V.
,
1986
, “
Asymptotic Analysis of the Lyapunov Exponents and Rotation Numbers of the Random Oscillator and Applications
,”
SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
,
46
, pp.
427
–
450
.
7.
Namachchivaya
,
N. S.
, and
Van Roessel
,
H. J.
,
1993
, “
Maximal Lyapunov Exponent and Rotation Numbers for Two Coupled Oscillators Drivern by Real Noise
,”
J. Stat. Phys.
,
71
, pp.
549
–
567
.
8.
Doyle
,
M. M.
, and
Namachchivaya
,
N. Sri.
,
1994
, “
Almost-Sure Asymptotic Stability of a General Four-Dimensional System Driven by Real Noise
,”
J. Stat. Phys.
,
75
, pp.
525
–
552
.
9.
Namachchivaya
,
N. S.
, and
Taiwar
,
S.
,
1993
, “
Maximal Lyapunov Exponent and Rotation Numbers for Stochastically Perturbed Co-dimension Two Bifurcation
,”
J. Sound Vib.
,
169
, pp.
349
–
372
.
10.
Gardiner, C. W., 1997, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd Ed., Springer, Berlin.
11.
Liu
,
X. B.
, and
Liew
,
K. M.
,
2003
, “
The Lyapunov Exponent for aCo-Dimension Two Bifurcation System Driven by a Real Noise
,”
Int. J. Non-Linear Mech.
,
38
, pp.
1495
–
1511
.
12.
Liu
,
X. B.
, and
Liew
,
K. M.
,
2004
, “
On the Almost-Sure Stability Condition for a Co-Dimensional Two Bifurcation System Under the Parametric Excitation of a Real Noise
,”
J. Sound Vib.
,
272
, pp.
85
–
107
.
13.
Roy
,
R. V.
,
1994
, “
Stochastic Averaging of Oscillators Excited by Coloured Gaussian Processes
,”
Int. J. Non-Linear Mech.
,
29
, pp.
461
–
475
.
14.
Karlin, S., and Taylor, H. M., 1981, A Second Course in Stochastic Processes, Academic Press, San Diego, CA.
15.
Liberzon
,
D.
, and
Brockett
,
R. W.
,
2000
, “
Spectral Analysis of Fokker-Planck and Related Operators Arising From Linear Stochastic Differential Equations
,”
SIAM J. Control Optim.
,
38
, pp.
1453
–
1467
.
16.
Guckenheimer, G., and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.
17.
Caughey, T. K., 1971, “Nonlinear Theory of Random Vibrations,” Advances in Applied Mechanics, Academic Press, New York, 11, pp. 209–253.
18.
Lin, Y. K., and Cai, G. Q., 1995, Probabilistic Structural Dynamics, Advanced Theory and Applications, McGraw-Hill, New York.
19.
Seaborn, J. B., 1991, Hypergeometric Functions and Their Applications, Springer-Verlag, New York.
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