Instability results for a Hill equation coupled with an asymmetrically nonlinear oscillator

We consider a Hill equation whose potential depends on the solution of a nonlinear oscillator. The nonlinearity of the oscillator is given by a function f(x) which has polynomial growth as x → +∞ and is asymptotically constant as x → −∞. We provide explicit conditions on a set of 4 parameters for the stability of the Hill equation as the energy of the oscillator approaches infinity. In the case when the ratio of the angular frequencies of the linearized system (around the null solution) is an integer, we recover the same instability intervals as in the case in which f was extended by symmetry to an odd function. When this ratio is not an integer, the system is essentially unstable at high energies. Finally, we consider the case where f has different polynomial growth orders to +∞ and to −∞, and generalize previous results of Cazenave and Weissler concerning the stability of a nonlinear mode of the Kirchhoff string equation. The problem and the choice of the assumptions on the function f are motivated by the (linear) stability analysis of a coupled nonlinear system of ODEs which is a simplified model for the interaction of flexural and torsional modes of vibration along the deck of a suspended bridge.