Bifurcations in a Mathieu equation with cubic nonlinearities

We investigate the nonlinear dynamics of the classical Mathieu equation to which is added a nonlinearity which is a general cubic in x, ẋ. We use a perturbation method (averaging) which is valid in the neighborhood of 2:1 resonance, and in the limit of small forcing and small nonlinearity. By comparing the predictions of first-order averaging with the results of numerical integration, we show that it is necessary to go to second-order averaging in order to obtain the correct qualitative behavior. Analysis of the resulting slow-flow equations is accomplished both analytically as well as by use of the software AUTO.