具有Duffing系数的Hill方程的共振舌和非线性梁方程的不稳定性
Resonance tongues for the Hill equation with Duffing coefficients and instabilities in a nonlinear beam equation
摘要：
考虑一类Hill方程，其中周期系数是某个Duffing方程的平方解加上一个常数.本文研究了Hill方程平凡解的稳定性，证明了Burdina（V.I. Burdina，Boundedness of solutions of a system of微分方程）对这种分析非常有帮助。在某些情况下，我们也能够确定精确解的雅可比椭圆函数。总的来说，我们得到了一个相当完整的图片的稳定和不稳定的地区。然后利用这些结果研究了某些梁方程中非线性模式的稳定性。
英文摘要：
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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