The van Kampen expansion for the Fokker-Planck equation of a duffing oscillator

In Rodriguez and van Kampen's 1976 paper a method of extracting information from the Fokker-Planck equation without having to solve the equation is outlined. The Fokker-Planck equation for a Duffing oscillator excited by white noise is expanded about the intensity α of the forcing function. This expansion is carried to order\(\mathcal{O}(\alpha ^{1/2} )\). However, no studies are made of the effects of the order of the expansion or variation of the parameters, nor are comparisons made to experimental results. In the present paper, the expansion is carried to a higher order,\(\mathcal{O}(\alpha ^{3/2} )\), results are presented and compared to Monte Carlo experiments using both white and colored noise, and parametric studies are performed on the intensity of the forcing function and the damping coefficient. It is found that the expansion method works well for the case of white noise and for colored noise where the correlation time is less than 0.1 sec, but fails to give certain details. It is also found that the system behaves as expected when the parameters are varied.