Advertisement

Some Recent Advances in Theory of Stochastically Excited Dissipative Hamiltonian Systems

  • W. Q. Zhu
Conference paper
Part of the Solid Mechanics and its Applications book series (SMIA, volume 47)

Abstract

Some recent advances in the theory of stochastically excited dissipative Hamiltonian systems made by the author and his co-workers are summarized. It is shown that the structure of the solution and the energy partition among various degrees of freedom of a stochastically excited dissipative Hamiltonian system depend upon the integrability and resonance of the Hamiltonian system modified by the Wong-Zakai correction terms. Three procedures, i. e., one for obtaining exact stationary solution, equivalent nonlinear system method and stochastic averaging method, for predicting the response of stochastically excited dissipative Hamiltonian systems are presented. It is pointed out that all presently available exact stationary solutions of nonlinear stochastic systems can be obtained by the present procedure as special cases and that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are included in the present stochastic averaging of quasi-Hamiltonian systems as two special cases.

Keywords

Hamiltonian System Integrable Hamiltonian System Stochastic Average Nonlinear Stochastic System Stochastic Average Method 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Binney, J.J. et al. (1992) The Theory of Critical Phenomena, An Introduction to the Renormalization Group, Clarendon Press, Oxford.zbMATHGoogle Scholar
  2. Cai, G.Q. and Lin, Y. K. (1988a) On exact stationary solutions of equivalent non-linear stochastic systems, Int. J. Non-Linear Mech. 23 315–325.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Cai, G.Q. and Lin, Y.K. (1988b) A new approximate solution technique for randomly excited non-linear oscillators, Int. J. Non-Linear Mech. 23 409–420.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Khasminskii, R. Z. (1964) On the behavior of a conservative system with friction and small random noise (in Russian). Prikladnaya Male matika i Mechanica (Appl. Math. Mech.), 28, 1126–1130.MathSciNetGoogle Scholar
  5. Khasminskii, R.Z. (1968) On the averaging principle for stochastic differential Itô equation (in Russian), Kibernetica 4, 260–279.Google Scholar
  6. Lutes, L.D. (1970) Approximate technique for treating random vibration of hysteretic systems, J. Acoust. Soc. Am. 48, 299–306.CrossRefGoogle Scholar
  7. Soize, C. (1988) Steady state solution of Fokker-Planck equation in higher dimension, Probabilistic Engineering Mechanics 3, 196–206.CrossRefGoogle Scholar
  8. Stratonovich, R. L. (1963) Topics in the Theory of Random Noise, Vol. 1, Gordon and Breach, New York.Google Scholar
  9. Tabor, M. (1989) Chaos and Integrability in Nonlinear Dynamics, John Wiley &.Sons, New York.zbMATHGoogle Scholar
  10. Zhu, W. Q., Cai, G. Q. and Lin, Y. K. (1990) On exact stationary solutions of stochasticlly perturbed Hamiltonian systems, Probabilistic Engineering Mechanics 5,84–89.CrossRefGoogle Scholar
  11. Zhu, W.Q. and Lei, Y. (1995) Equivalent nonlinear system method for stochastically excited dissipative integrable Hamiltonian systems, submitted to ASME J. Appl. Mech. Google Scholar
  12. Zhu, W.Q., Soong, T. T. and Lei, Y. (1994) Equivalent nonlinear system method for stochastically excited Hamiltonian systems, ASME, J. Appl. Mech. 61, 618–623.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Zhu, W. Q. and Yang,Y. Q. (1995a) Exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems, to appear in Asme.J. Appl. Mech. Google Scholar
  14. Zhu, W.Q. and Yang, Y.Q. (1995b) Stochastic averaging of quasi-nonintegrable-Hamil- tonian systems, submitted to ASME J. Appl. Mech. Google Scholar
  15. Zhu, W.Q. and Yang, Y.Q. (1995c) Stochastic averaging of quasi-integrable Hamil-tonian systems, submitted to ASME J. Appl. Mech. Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • W. Q. Zhu
    • 1
  1. 1.Department of MechanicsZhejiang UniversityHangzhouP. R. China

Personalised recommendations