Floquet乘子与周期线性微分方程的稳定性：统一算法及其计算机实现
Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization
摘要：
Floquet乘子（特征乘子）对周期方程的稳定性起着重要作用。基于迭代方法，我们提供了一个统一的算法来计算Floquet乘子（特征乘子）和确定时间尺度上的周期线性微分方程的稳定性，从而统一了离散、连续和混合动力学。我们的方法是基于计算A和B的值（见定理3.1），它们分别是系统的所有Floquet乘数（特征乘数）的总和和乘积。我们用变分法和逼近论得到了A的显式表达式（见定理4.1），用刘维公式得到了B的显式表达式。最后，设计了一个计算机程序来实现我们的算法。具体来说，您可以确定二阶周期性线性系的稳定性，无论它们是离散的、连续的还是混合的，只要您输入与方程参数相关联的程序代码。实际上，关于Floquet乘子的计算方法的文献很少，更没有设计计算机实现的程序。我们的算法给出了所有Floquet乘子的显式表达式，我们的计算机程序是基于这些显式表达式的近似。特别地，在任意离散周期时间尺度上，我们可以进行有限次计算以得到Floquet乘数的显式值（见定理4.2）。因此，对于任何离散周期系统，即使不需要计算机，我们也可以准确地判定系统的稳定性!最后，在第6节中，给出了几个例子来说明我们的算法的有效性。
英文摘要：
Floquet multipliers (characteristic multipliers) play significant role in the stability of the periodic equations. Based on the iterative method, we provide a unified algorithm to compute the Floquet multipliers (characteristic multipliers) and determine the stability of the periodic linear differential equations on time scales unifying discrete, continuous, and hybrid dynamics. Our approach is based on calculating the value of A and B (see Theorem 3.1), which are the sum and product of all Floquet multipliers (characteristic multipliers) of the system, respectively. We obtain an explicit expression of A (see Theorem 4.1) by the method of variation and approximation theory and an explicit expression of B by Liouville's formula. Furthermore, a computer program is designed to realize our algorithm. Specifically, you can determine the stability of a second order periodic linear system, whether they are discrete, continuous or hybrid, as long as you enter the program codes associated with the parameters of the equation. In fact, few literatures have dealt with the algorithm to compute the Floquet multipliers, not mention to design the program for its computer realization. Our algorithm gives the explicit expressions of all Floquet multipliers and our computer program is based on the approximations of these explicit expressions. In particular, on an arbitrary discrete periodic time scale, we can do a finite number of calculations to get the explicit value of Floquet multipliers (see Theorem 4.2). Therefore, for any discrete periodic system, we can accurately determine the stability of the system even without computer! Finally, in Section 6, several examples are presented to illustrate the effectiveness of our algorithm.
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