Stochastic Averaging of Quasi-Integrable Hamiltonian Systems

[+] Author and Article Information
W. Q. Zhu, Z. L. Huang, Y. Q. Yang

Department of Mechanics, Zhejiang University, Hangzhou 310027, R. R. China

J. Appl. Mech 64(4), 975-984 (Dec 01, 1997) (10 pages) doi:10.1115/1.2789009 History: Received June 12, 1995; Revised April 09, 1997; Online October 25, 2007


A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems, i.e., multi-degree-of-freedom integrable Hamiltonian systems subject to lightly linear and (or) nonlinear dampings and weakly external and (or) parametric excitations of Gaussian white noises. According to the present method an n-dimensional averaged Fokker-Planck-Kolmogrov (FPK) equation governing the transition probability density of n action variables or n independent integrals of motion can be constructed in nonresonant case. In a resonant case with α resonant relations, an (n + α)-dimensional averaged FPK equation governing the transition probability density of n action variables and α combinations of phase angles can be obtained. The procedures for obtaining the stationary solutions of the averaged FPK equations for both resonant and nonresonant cases are presented. It is pointed out that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are two special cases of the present stochastic averaging. Two examples are given to illustrate the application and validity of the proposed method.

Copyright © 1997 by The American Society of Mechanical Engineers
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