Dynamics of a nonlinear parametrically excited partial differential equation

We investigate a parametrically excited nonlinear Mathieu equation with dam ping and limited spatial dependence, using both perturbation theory and num erical integration. The perturbation results predict that, for parameters w hich lie near the 2:1 resonance tongue of instability corresponding to a si ngle mode of shape cos nx, the resonant mode achieves a stable periodic mot ion, while all the other modes are predicted to decay to zero. By numerical ly integrating the p.d.e. as well as a 3-mode o.d.e. truncation, the predic tions of perturbation theory are shown to represent an oversimplified pictu re of the dynamics. In particular it is shown that steady states exist whic h involve many modes. The dependence of steady state behavior on parameter values and initial conditions is investigated numerically. (C) 1999 America n Institute of Physics. [S1054-1500(99)00601-1].