The van Kampen expansion for the Fokker-Planck equation of a duffing oscillator excited by colored noise

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. In Weinstein and Benaroya, the effect of the order of expansion is investigated by carrying the expansion to a higher order. The effect of varying the system parameters is also investigated. All results are verified by comparison to Monte Carlo experiments. In this paper, the van Kampen expansion is modified and applied to the case of a Duffing oscillator excited by colored noise. The effect of the correlation time is investigated. Again the results are compared to those of Monte Carlo experiments. It is found that the expansion compares closely with those of the Monte Carlo experiments as the correlation time τc is varied from 0.001 to 10 sec. Examination of the results reveals that the colored noise can be categorized in one of four ways: (1) for\(\tau _c< \mathcal{O}(0.01{\text{ }}\sec )\) the noise can be considered as white for all intents and purposes, (2) for\(\tau _c = \mathcal{O}(0.1{\text{ }}\sec )\) the noise can be considered white for some purposes, (3) for\(\tau _c = \mathcal{O}(1.0{\text{ }}\sec )\) the correlated nature of the noise must be considered in an analysis, and (4) for\(\mathcal{O}(1.0{\text{ }}\sec )< \tau _c \) the noise can be considered as deterministic.