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

## TL;DR

This paper analyzes the stability of solutions in a class of Hill equations with Duffing-based coefficients, using Burdina's criterion and elliptic functions, and applies findings to nonlinear beam equations.

## Contribution

It introduces a detailed stability analysis for Hill equations with Duffing coefficients and connects these results to nonlinear beam equation instabilities.

## Key findings

- Complete characterization of stability and instability regions.
- Application of Burdina's criterion for stability analysis.
- Exact solutions expressed via Jacobi elliptic functions.

## 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.