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A New Tool for the Investigation of a Class of Nonlinear Stochastic Differential Equations: the Melnikov Process

  • E. Simiu
  • M. Franaszek
Conference paper
Part of the Solid Mechanics and its Applications book series (SMIA, volume 47)

Abstract

The Melnikov process, a construct rooted in chaotic dynamics theory, was recently developed as a tool for the investigation of a broad class of nonlinear stochastic differential equations [1–6]. This paper briefly reviews the stochastic Melnikov-based approach and applications to (i) oceanography, (ii) open-loop control of stochastic nonlinear systems, and (iii) snap-through of buckled beams with distributed mass and distributed random loading.

Keywords

Control Force Unstable Manifold Homoclinic Orbit Escape Rate Melnikov Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Frey, M. and Simiu E., “Noise-induced chaos and phase space flux,” Physica D 63, 321-340,1993.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Hsieh,, S.R., Troesch, A.W., and Shaw, S.W., “A nonlinear probabilistic method for predicting vessel capsizing in random beam seas,” Proc. Royal Soc. London A 446, pp. 195-211, 1994.CrossRefGoogle Scholar
  3. 3.
    Simiu, E., “Melnikov Process for Stochastically Perturbed Slowly Vaiying Oscillator: Application to a Model of Wind-driven Coastal Currents,” J. Appl. Mech. (in press).Google Scholar
  4. 4.
    Simiu, E. and Frey, M., “Melnikov Processes and Noise-induced Exits from a Well,” J.Eng.Mech., Feb. 1996 (in press).Google Scholar
  5. 5.
    Simiu, E., and Hagwood, C., “Exits in Second-Order Nonlinear Systems Driven by Dichotomous Noise” Proc., 2nd Int. Conf. Comp. Stock. Mech. (P. Spanos, ed.), pp. 395-401, Balkema, 1995.Google Scholar
  6. 6.
    Franaszek, M., and Simiu, E., “Noise-induced Snap-through of Buckled Column With Continuously Distributed Mass: A Chaotic Dynamics Approach,” submitted to Int. J. Non-linear Mech. Google Scholar
  7. 7.
    Shinozuka, M “Simulation of Multivariate and Multidimensional Random Processes, J. AcousL Soc. Amer. 49, 347-357, 1971.Google Scholar
  8. 8.
    Wiggins, S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos, New York: Springer-Verlag.zbMATHGoogle Scholar
  9. 9.
    Moon, F.C., Chaotic Vibrations, New York: John Wiley and Sons, 1987.zbMATHGoogle Scholar
  10. 10.
    Holmes, P. and Marsden, J., “A Partial Differential Equation with Infinitely Many Periodic Orbits: Chaotic Oscillations of a Forced Beam,” Arch. Rat Mech. Analys. 76 135-166,1985.MathSciNetGoogle Scholar
  11. 11.
    Allen, J.S., Samelson, R.M. and Newberger, P.A, “Chaos in a Model of Forced Quai-geostrophic Flow Over Topography: an Application of Melnikov’s Method,” J. Fluid Mech., 226, 511-547,1991.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Wiggins, S. and Holmes, P., “Homoclinic Orbits in Slowly Vaiying Oscillators,” SIAM J. Math. Anal. 18 612-629; Errata: 19 1254-1255,1987.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    E. Simiu and M. Franaszek, “Melnikov-based Open-loop Control of Escape for a Class of Nonlinear Systems,” Proc., Symp. on Vibr. Contr. Stoch. Dynam. Syst., ASME, (L. Bergman, ed.), Sept. 1995.Google Scholar
  14. 14.
    Tseng, W.Y. and Dugundji, J., “Nonlinear Vibrations of a Buckled Beam Under Harmonic Excitation,” J. AppL Mech. 467-476, 1971.Google Scholar
  15. 15.
    Meirovich, L., Analytical Methods in Vibration, Elsevier, New York, 1964.Google Scholar
  16. 16.
    Sivathanu, Y., Hagwood, C., and Simiu, E., “Exits in multistable systems excited by coin-toss square wave dichotomous noise: a chaotic dynamics approach,” Phys. Rev. E (to be published) Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • E. Simiu
    • 1
  • M. Franaszek
    • 1
  1. 1.National Institute of Standards and TechnologyGaithersburgUSA

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