The stochastic Melnikov approach is extended to a class of slowly varying dynamical systems. It is found that (1) necessary conditions for chaos induced by stochastic perturbations depend on the excitation spectrum and the transfer function in the expression for the Melnikov transform; (2) the Melnikov approach allows the estimation of lower bounds for (a) the mean time of exit from preferred regions of phase space, and (b) the probability that exits from those regions cannot occur during a specified time interval. For a system modeling wind-induced currents, the deterministic Melnikov approach would indicate that chaotic transport cannot occur for certain parameter ranges. However, the more realistic stochastic Melnikov approach shows that, for those same parameter ranges, the necessary conditions for exits during a specified time interval are satisfied with probabilities that increase as the time interval increases.

1.
Allen
J. S.
,
Samelson
R. M.
, and
Newberger
P. A.
,
1991
, “
Chaos in a Model of Forced Quasi-Geostrophic Flow Over Topography: An Application of Melnikov’s Method
,”
Journal of Fluid Mechanics
, Vol.
226
, pp.
511
547
.
2.
Berge´, P., Pomeau, Y., and Vidal, C., 1984, Order Within Chaos, John Wiley and Sons, New York.
3.
Beigie
D.
,
Leonard
A.
, and
Wiggins
S.
,
1991
, “
Chaotic Transport in the Homoclinic and Heteroclinic Tangle Regions of Quasiperiodically Forced Two-Dimensional Dynamical Systems
,”
Nonlinearity
, Vol.
4
, pp.
775
819
.
4.
Frey, M., and Simiu, E., 1993, “Noise-Induced Chaos and Phase Space Flux,” Physica D, pp. 321–340.
5.
Frey, M., and Simiu, E., 1995, “Noise-Induced Transitions to Chaos,” Proceedings, NATO Advanced Workshop on Spatio-Temporal Patterns in Nonequilibrium Complex Systems, P. Cladis and P. Pally-Muhoray, eds., Addison-Wesley, New York, pp. 529–544.
6.
Hsieh
S. R.
,
Troesch
A. W.
, and
Shaw
S.
,
1994
, “
A Nonlinear Probabilistic Method for Predicting Vessel Capsizing in Random Beam Seas
,”
Proceedings, Royal Society of London
, Vol.
A446
, pp.
195
211
.
7.
NOAA, 1977, “Local Climatological Annual Summaries,” National Oceanic and Atmospheric Administration, Environmental Data Service, Asheville, N.C.
8.
Rice, S. O., 1954, “Mathematical Analysis of Random Noise,” Selected Papers in Noise and Stochastic Processes, A. Wax, ed., Dover, New York.
9.
Shinozuka
M.
,
1971
, “
Simulation of Multivariate and Multidimensional Random Processes
,”
Journal of the Acoustical Society of America
, Vol.
49
, pp.
347
357
.
10.
Shinozuka
M.
, and
Deodatis
G.
,
1991
, “
Simulation of Stochastic Processes by Spectral Representation
,”
ASME Applied Mechanics Reviews
, Vol.
44
, pp.
191
204
.
11.
Simiu, E., and Scanlan, R. H., 1986, Wind Effects on Structures, Second Edition, John Wiley and Sons, New York.
12.
Simiu, E., and Frey, M., 1996, “Noise-Induced Sensitivity to Initial Conditions,” Proceedings, Workshop on Fluctuations and Order, M. Millonas, ed., Springer-Verlag, New York, pp. 81–90.
13.
Simiu, E., and Hagwood, C., 1994, “Exits in Second-Order Nonlinear Systems Driven by Dichotomous Noise,” Proceedings, Conference on Computational Stochastic Mechanics, P. Spanos and M. Shinozuka, eds., Athens, Sept. (in press).
14.
Vaicaitis
R.
, and
Simiu
E.
,
1977
, “
Nonlinear Pressure Terms and Along-Wind Response
,”
Journal of the Structural Division, ASCE
, Vol.
103
, pp.
903
906
.
15.
Verhulst, F., 1990, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York.
16.
Van der Hoven
I.
,
1957
, “
Power Spectrum of Horizontal Wind Speed in the Frequency Range from 0.0007 to 900 Cycles per Hour
,”
Journal of Meteorology
, Vol.
14
, pp.
160
163
.
1.
Wiggins
S.
, and
Holmes
P.
,
1987
, “
Homoclinic Orbits in Slowly Varying Oscillators
,”
SIAM Journal for Mathematical Analysis
, Vol.
18
, pp.
612
629
;
2.
Errata
,
SIAM Journal for Mathematical Analysis
, Vol.
19
, pp.
1254
1255
.
1.
Wiggins
S.
, and
Shaw
S. W.
,
1988
, “
Chaos and Three-Dimensional Horseshoes in Slowly Varying Oscillators
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
55
, pp.
959
968
.
2.
Yim, S. C. S., and Lin, H., 1992, “Probabilistic Analysis of a Chaotic Dynamical System,” Applied Chaos, J. H. Kim and J. Stinger, eds., John Wiley and Sons, New York.
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