Abstract
We have developed a generalization of the method of statistical linearization to enable us to describe transient and other nonstationary phenomena obeying stochastic nonlinear differential equations. This approximation technique provides an optimal Gaussian representation with time-dependent parameters. The algorithm specifies a set of ordinary differential equations for the Gaussian parameters in terms of the time-dependent average nonlinearities. We apply the general formalism developed herein for single degree of freedom dissipative systems to a particular example.
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References
A. Schenzle and H. Brand, Multiplicative stochastic processes in statistical physics,Phys. Rev. A 20:1628–1647 (1979).
K. Lindenberg, K. E. Shuler, V. Seshadri, an B. J. West, inProbabilistic Analysis and Related Topics, A. T. Bharucha-reid, ed. (Academic Press, New York, 1983).
Z. Schuss,Theory and Applications of Stochastic Differential Equations (John Wiley, New York, 1980).
J. D. Mason, ed.,Stochastic Differential Equations and Applications. (Academic Press, New York, 1977).
R. C. Booton, The analysis of nonlinear control systems with random inputs,IRE Trans. Circuit Theory 1:32–34 (1954).
I. E. Kazakov, Approximate probabilistic analysis of the accuracy of performance of essentially nonlinear automatic systems.Automatika i Telemekanika 17:385–409 (1956).
T. K. Caughey, Equivalent linearization techniques.J. Acoust. Soc. Am. 35:1706–1711 (1963).
T. K. Caughey, inAdvances in Applied Mechanics Vol. 11, pp. 209–253. (Academic Press, New York, 1971).
S. H. Crandall, inProblems of the Asymptotic Theory of Nonlinear Oscillations, Mitropolsky Anniversary Volume, (Naukova Dumka, Kiev, 1977).
S. H. Crandall, Non-Gaussian closure for random vibration of non-linear oscillators.Int. J. Nonlinear Mech. 15:303–314 (1980).
A. B. Budgor, K. Lindenberg, and K. E. Shuler, Studies in non-linear stochastic processes. II. The Duffing oscillator revisited.J. Stat. Phys. 15:375–391 (1976).
W. D. Iwan and I-Min Yang, Application of statistical linearization techniques to nonlinear multidegree-of-freedom systems.J. Appl. Mech. 39:545–550 (1972).
W. D. Iwan, A generalization of the concept of equivalent linearization.Int. J. Nonlinear Mech. 8:279–287 (1973).
T. S. Atalik and S. Utku, Stochastic linearization of multidegree-of-freedom nonlinear systems.Earthquake Eng. Struct. Dyn. 4:411–420 (1976).
B. J. West, K. Lindenberg, and K. E. Schuler, Studies in nonlinear stochastic processes. IV. A comparison of statistical linearization, diagrammatic expansion, and projection operator methods.J. Stat. Phys. 18:217–233 (1978).
B. J. West, Statistical properties of water waves. I. Steady-state distribution of wind-driven gravity-capillary waves.J. Fluid Mech. 117:187–210 (1982).
F. de Pasquale, P. Tartaglia, and P. Tombesi, New expansion technique for the decay of an unstable state.Phys. Rev. A 25:466–471 (1982).
A. R. Bulsara, K. Lindenberg, and K. E. Shuler, Spectral analysis of a nonlinear oscillator driven by random and periodic forces. I. Linearized theory.J. Stat. Phys. 27:787–808 (1982).
E. Ben-Jacob, D. S. Bergman, and Z. Schuss, Thermal fluctuations and lifetime of the nonequilibrium steady state in a hysteretic Josephson junction.Phys. Rev. B 25:519–522 (1982).
J. O. Eaves and W. P. Rinehardt, Piecewise optimal linearization method for nonlinear stochastic differential equations.J. Stat. Phys. 25:127–141 (1981).
W. D. Iwan and A. B. Mason, Equivalent linearization for systems subjected to nonstationary random excitations.Int. J. Nonlinear Mech. 15:71–81 (1980).
M. Lax, Classical noise. III. Nonlinear Markoff processes.Rev. Mod. Phys. 38:359–566 (1966).
K. Lindenberg, B. J. West, and G. Rovner, in preparation.
M. C. Valsakumar, K. P. N. Murthy, and G. Ananthakrishna,J. Stat. Phys. 30:617–631 (1983).
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Supported by the Defense Advanced Research Projects Agency DARPA-0053.
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West, B.J., Rovner, G. & Lindenberg, K. Approximate Gaussian representation of evolution equations I. Single degree of freedom nonlinear equations. J Stat Phys 30, 633–648 (1983). https://doi.org/10.1007/BF01009681
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DOI: https://doi.org/10.1007/BF01009681