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TECHNICAL PAPERS

Moment Lyapunov Exponent and Stochastic Stability of Two Coupled Oscillators Driven by Real Noise

[+] Author and Article Information
N. Sri Namachchivaya

Department of Aeronautical and Astronautical Engineering, University of Illinois, 103 S. Wright Street, Urbana, IL 61801-2935

H. J. Van Roessel

Department of Mathematical Sciences, University of Alberta, Edmonton AB, Canada

J. Appl. Mech 68(6), 903-914 (Feb 26, 2001) (12 pages) doi:10.1115/1.1387021 History: Received May 15, 2000; Revised February 26, 2001
Copyright © 2001 by ASME
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References

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Arnold,  L., 1984, “A Formula Connecting Sample and Moment Stability of Linear Stochastic Systems,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 44, No. 4, pp. 793–802.
Arnold, L., Kliemann, W., and Oeljeklaus, E., 1986, Lyapunov Exponents of Linear Stochastic Systems, Vol. 1186 (Lecture Notes in Mathematics), Springer-Verlag, New York, pp. 85–125.
Arnold, L., Oeljeklaus, E., and Pardoux, E., 1986, Almost Sure and Moment Stability for Linear Ito⁁ Equations, Vol. 1186 (Lecture Notes in Mathematics), Springer-Verlag, New York, pp. 129–159.
Namachchivaya,  N. Sri, Van Roessel,  H. J., and Doyle,  M. M., 1996, “Moment Lyapunov Exponent for Two Coupled Oscillators Driven by Real Noise,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 56, pp. 1400–1423.
Khas’minskii,  R. Z., and Moshchuk,  N., 1998, “Moment Lyaponov 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.
Arnold,  L., Doyle,  M. M., and Namachchivaya,  N. Sri, 1997, “Small Noise Expansion of Moment Lyapunov Exponents for General Two Dimensional Systems,” Dyn. Stab. Syst., 12, No. 3, pp. 187–211.
Pardoux,  E., and Wihstutz,  V., 1988, “Lyapunov Exponent and Rotation Number of Two-Dimensional Linear Stochastic Systems With Small Diffusion,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 48, No. 2, pp. 442–457.
Namachchivaya,  N. Sri, and Van Roessel,  H. J., 1993, “Maximal Lyapunov Exponent and Rotation Numbers for Two Coupled Oscillators Driven by Real Noise,” J. Stat. Phys., 71, No. 3/4, pp. 549–567.
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Wedig, W. V., 1988, “Lyapunov Exponents of Stochastic Systems and Related Bifurcation Problems,” Stochastic Structural Dynamics: Progress in Theory and Applications, S. T. Ariaratnam, G. I. Schuëller, and I. Elishakoff, eds., Elsevier, London.
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Figures

Grahic Jump Location
Moment Lyapunov exponent for the general case S(2ω1)=1,S(2ω2)=2,S(ω12)=2,S(ω1−ω2)=1,p11=1,p22=2,p12=p21=κ=1,δ1=1,δ2=2
Grahic Jump Location
Moment Lyapunov exponent for the general case S(2ω1)=1,S(2ω2)=2,S(ω12)=2,S(ω1−ω2)=1,p11=1,p22=2,p12=−p21=κ=1,δ1=1,δ2=2
Grahic Jump Location
Thin rectangular beam subjected to stochastic excitation
Grahic Jump Location
Almost-sure stability boundary for follower force and end moment cases under real noise excitation, i.e., ω1=0.5,ω2=2, κ=1, δ1=.001,delta2=.002, α=0.5
Grahic Jump Location
Moment Lyapunov exponent for the follower force problem with κ=1, δ1=0.5,δ2=1,S(ω+)=1,S(ω)=1

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