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TECHNICAL PAPERS

Lyapunov Exponents and Moment Lyapunov Exponents of a Two-Dimensional Near-Nilpotent System

[+] Author and Article Information
W.-C. Xie

Solid Mechanics Division, Faculty of Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada

J. Appl. Mech 68(3), 453-461 (Nov 08, 2000) (9 pages) doi:10.1115/1.1364491 History: Received November 09, 1999; Revised November 08, 2000
Copyright © 2001 by ASME
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References

Horsthemke, W., and Lefever, R., 1984, Noise-Induced Transitions, Springer-Verlag, Berlin.
Baxendale, P. H., 1991, “Invariant Measure for Nonlinear Stochastic Differential Equations,” Lyapunov Exponents (Lecture Notes in Mathematics, 1486), L. Arnold, H. Crauel, and J.-P. Eckmann, eds., Springer-Verlag, Berlin, pp. 123–140.
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. 793–801.
Arnold, L., 1998, Random Dynamical Systems, Springer-Verlag, Berlin, Chapter 9.
Khasminskii,  R., 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.
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.
Ariaratnam,  S. T., and Xie,  W.-C., 1993, “Lyapunov Exponents and Stochastic Stability of Two-Dimensional Parametrically Excited Random Systems,” Trans. ASME J. Appl. Mech., 60, pp. 677–682.
Khasminskii,  R. Z., 1967, “Necessary and Sufficient Conditions for the Asymptotic Stability of Linear Stochastic Systems,” Theor. Probab. Appl., 12, pp. 144–147 (English translation).
Milstein,  G. N., 1996, “Evaluation of Moment Lyapunov Exponents for Second Order Stochastic Systems,” Random Comput. Dyn., 4, No. 4, pp. 301–315.
Ariaratnam,  S. T., and Xie,  W.-C., 1990, “Lyapunov Exponent and Rotation Number of a Two-Dimensional Nilpotent Stochastic System,” Dyn. Stab. Syst.,5, No. 1, pp. 1–9.
Wedig, W., 1988, “Lyapunov Exponent 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, New York, pp. 315–327.
Wedig, W., 1995, “Pitchfork and Hopf Bifurcations in Stochastic Systems—Effective Methods to Calculate Lyapunov Exponents,” Probabilistic Methods in Applied Physics, P. Krée and W. Wedig, eds., Springer-Verlag, Berlin, pp. 120–148.
Baxendale,  P., and Stroock,  D., 1988, “Large Deviations and Stochastic Flows of Diffeomorphisms,” Prob. Theory Related Fields,80, pp. 169–215.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., 1992, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed., Cambridge University Press, Cambridge, UK.

Figures

Grahic Jump Location
Points of dynamic bifurcation
Grahic Jump Location
Points of dynamic bifurcation
Grahic Jump Location
Moment Lyapunov exponents of the nilpotent system
Grahic Jump Location
Moment Lyapunov exponents of the nilpotent system
Grahic Jump Location
Moment Lyapunov exponents  
Grahic Jump Location
Moment Lyapunov exponents  

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