标题： 带有随机强迫项的希尔方程 显示英文标题

标题： Hill's Equation with Random Forcing Terms

摘要： 受天文学中一类轨道问题的启发，本文研究了希尔方程的解，其中强迫强度参数在每个周期内发生变化。结果被推广到包括周期在每个周期内的变化。微分方程解的发展由一个离散映射控制。对于希尔方程在不稳定极限下的普遍情况，我们分别考虑矩阵元素仅为正值和具有混合符号的情况；然后我们找到了增长速率的确切表达式、界限和工作估计。我们还找到了若干$2 \times 2$矩阵的无穷乘积的确切表达式、估计值和界限，这些矩阵的矩阵元素中含有随机变量。在尖峰强迫项（δ函数极限）的情况下，我们为每个周期以及匹配周期间解的离散映射找到解析解；对于这种情况，我们在大强迫强度的极限下找到增长速率和不稳定性条件，以及稳定/不稳定区域的宽度。 显示英文摘要

摘要： Motivated by a class of orbit problems in astrophysics, this paper considers solutions to Hill's equation with forcing strength parameters that vary from cycle to cycle. The results are generalized to include period variations from cycle to cycle. The development of the solutions to the differential equation is governed by a discrete map. For the general case of Hill's equation in the unstable limit, we consider separately the case of purely positive matrix elements and those with mixed signs; we then find exact expressions, bounds, and working estimates for the growth rates. We also find exact expressions, estimates, and bounds for the infinite products of several $2 \times 2$ matrices with random variables in the matrix elements. In the limit of sharply spiked forcing terms (the delta function limit), we find analytic solutions for each cycle and for the discrete map that matches solutions from cycle to cycle; for this case we find the growth rates and the condition for instability in the limit of large forcing strength, as well as the widths of the stable/unstable zones.