On double-peak probability density functions of a Duffing oscillator under narrow-band random excitations

The principal resonance of Duffing oscillator under the narrow-band random external excitation is investigated. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The one-peak probability density function of each of the two stable stationary solutions is then calculated by perturbation analysis in the resonant case. These two functions are combined by using the probability of realization of the two stable stationary solutions to calculate the double-peak probability density function approximately. Theoretical analyses are verified by numerical results.