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.
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References
Frey, M. and Simiu E., “Noise-induced chaos and phase space flux,” Physica D 63, 321-340,1993.
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.
Simiu, E., “Melnikov Process for Stochastically Perturbed Slowly Vaiying Oscillator: Application to a Model of Wind-driven Coastal Currents,” J. Appl. Mech. (in press).
Simiu, E. and Frey, M., “Melnikov Processes and Noise-induced Exits from a Well,” J.Eng.Mech., Feb. 1996 (in press).
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.
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.
Shinozuka, M “Simulation of Multivariate and Multidimensional Random Processes, J. AcousL Soc. Amer. 49, 347-357, 1971.
Wiggins, S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos, New York: Springer-Verlag.
Moon, F.C., Chaotic Vibrations, New York: John Wiley and Sons, 1987.
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.
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.
Wiggins, S. and Holmes, P., “Homoclinic Orbits in Slowly Vaiying Oscillators,” SIAM J. Math. Anal. 18 612-629; Errata: 19 1254-1255,1987.
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.
Tseng, W.Y. and Dugundji, J., “Nonlinear Vibrations of a Buckled Beam Under Harmonic Excitation,” J. AppL Mech. 467-476, 1971.
Meirovich, L., Analytical Methods in Vibration, Elsevier, New York, 1964.
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)
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© 1996 Kluwer Academic Publishers
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Simiu, E., Franaszek, M. (1996). A New Tool for the Investigation of a Class of Nonlinear Stochastic Differential Equations: the Melnikov Process. In: Naess, A., Krenk, S. (eds) IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics. Solid Mechanics and its Applications, vol 47. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0321-0_38
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DOI: https://doi.org/10.1007/978-94-009-0321-0_38
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