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ABSTRACT. The finite element method has been previously applied to the solution of the Fokker-Planck equation for second order systems under additive white noise excitation. Animation of these solutions has provided insights into previously unstudied nonstationary behavior of the transition probability density functions of these systems. This work has now been extended to encompass systems subjected to both additive and multiplicative white noise excitations. In this paper we will examine a sequence of Duffing systems having a negative linear restoring force, and we will demonstrate through inspection of the probability density functions that adding parametric excitation to the linear part of the stiffness term results in stabilization at the origin. We will also compute second moments for each system and show the limited accuracy of several closure methods for this class of systems.(please replace with the abstract of your paper)

 

 

REFERENCES

 

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3. Kloeden, P. E. and Platen, E. , (1992). Numerical Solution of Stochastic Di_erential Equations, Springer-Verlag, Berlin,.

4. Langley, R. S. , (1985).A Finite Element Method for the Statistics of Non-Linear Random Vibration, Journal of Sound and Vibration 101 , 41-54.

5. Langtangen, H. P. , (1991). A General Numerical Solution Method for Fokker-Planck Equations with Applications to Structural Reliability, Probabilistic Engineering Mechanics 6 , 33-48.

6. Namachchivaya, N. Sri , (1990). Stochastic Bifurcation, Applied Mathematics and Computation 38 , Elsevier Science Publishing Co., Inc., 101-159.

7. Spencer, B. F. , Jr. and Bergman, L. A. , (1993). On the Numerical Solution of the Fokker-Planck Equation for Nonlinear Stochastic Systems, Nonlinear Dynamics 4, Kluwer Academic Publishers., 357-372.(please replace with the references of your paper)