We consider a class of Hill equations where the periodic coefficient is thesquared solution of some Duffing equation plus a constant. We study thestability of the trivial solution of this Hill equation and we show that acriterion due to Burdina (V.I. Burdina, Boundedness of solutions of a system ofdifferential equations) is very helpful for this analysis. In some cases, weare also able to determine exact solutions in terms of Jacobi ellipticfunctions. Overall, we obtain a fairly complete picture of the stability andinstability regions. These results are then used to study the stability ofnonlinear modes in some beam equations.
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