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arXiv · 1703.06362

Resonance tongues for the Hill equation with Duffing coefficients and instabilities in a nonlinear beam equation

Carlo Gasparetto·Filippo Gazzola
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Abstract

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

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Carlo Gasparetto, Filippo Gazzola. 2017-03-18. Resonance tongues for the Hill equation with Duffing coefficients and instabilities in a nonlinear beam equation. https://doi.org/10.1142/s0219199717500225

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