# On the instability tongues of the Hill equation coupled with a conservative nonlinear oscillator

## TL;DR

This paper analyzes the asymptotic behavior of instability regions in Hill equations coupled with nonlinear oscillators, revealing Mathieu-like behavior at small energies and characterizing tongue shapes in key cases.

## Contribution

It provides a detailed asymptotic analysis of instability tongues for coupled Hill equations, extending Mathieu equation results to more complex systems.

## Key findings

- Instability tongues have length $O(q^N)$ as $q o 0$
- Characterization of tongue shapes for small energies in specific cases
- Full class of Hill equations with similar asymptotic behavior identified

## Abstract

We study the asymptotics for the lengths $L_N(q)$ of the instability tongues of Hill equations that arise as iso-energetic linearization of two coupled oscillators around a single-mode periodic orbit. We show that for small energies, i.e. $q\rightarrow 0$, the instability tongues have the same behavior that occurs in the case of the Mathieu equation: $L_N(q) = O(q^N)$. The result follows from a theorem which fully characterizes the class of Hill equations with the same asymptotic behavior. In addition, in some significant cases we characterize the shape of the instability tongues for small energies. Motivation of the paper stems from recent mathematical works on the theory of suspension bridges.