Nonlinear oscillations under multifrequency parametric excitation

A second-order system of differential equations containing a multifrequency parametric excitation and weak quadratic and cubic nonlinearities is investigated. The method of multiple scales is used to carry out a general analysis, and three resonance conditions are considered in detail. First, the case in which the sum of two excitation frequencies is near two times a natural frequency, λ_s + λ_t ~2Ï _q, is examined. Second, the influence of an internal resonance, Ï _{q</sub =~3Ï r, on the previous case is studied. Finally, the effect of the internal resonance wr~3wq on the resonance λs + λt ~2Ï q is investigated. Results are presented as plots of response amplitudes as functions of a detuning parameter, excitation amplitude, and, for the first case, a measure of the relative values of λs + λt.}