Nonlinear Effects on Coexistence Phenomenon in Parametric Excitation

We investigate the effect of nonlinearites on a parametrically excited ordinary differential equation whose linearization exhibits the phenomenon of coexistence. The differential equation studied governs the stability of a mode of vibration in an unforced conservative two degree of freedom system used to model thefree vibrations of a thin elastica. Using perturbation methods, we show thatat parameter values corresponding to coexistence, nonlinear terms can cause the origin to become nonlinearly unstable,even though linear stability analysis predicts the origin to be stable.We also investigate the bifurcations associated with this instability.