耦合保守非线性振子的Hill方程的不稳定舌
On the instability tongues of the Hill equation coupled with a conservative nonlinear oscillator
摘要：
我们研究了两个耦合振子在单模周期轨道上等能线性化时Hill方程不稳定舌长度$L_N(q)$的渐近性。我们表明，对于小的能量，即$q\rightarrow 0$，不稳定舌有相同的行为，发生在马蒂厄方程的情况下：$L_N(q) = O(q^N)$。结果如下定理充分刻画类希尔方程具有相同的渐近行为。此外，在一些重要的情况下，我们描述的形状的不稳定舌小的能量。动机的文件源于最近的数学工作的理论悬索桥。
英文摘要：
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.
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