We study the asymptotics for the lengths of the instability tongues L_N(q) 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->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.

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

C. Marchionna;
2019-01-01

Abstract

We study the asymptotics for the lengths of the instability tongues L_N(q) 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->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.
2019
Hill equation,.Mathieu equation. Instability tongues. Coupled oscillators. Coexistence
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1099137
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