论文标题
Floquet乘数和周期性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
论文作者
论文摘要
Floquet乘数（特征乘数）在周期方程的稳定性中起着重要作用。基于迭代方法，我们提供了一种统一的算法来计算Floquet乘数（特征乘数），并在时间尺度上确定周期性线性微分方程的稳定性，以统一离散，连续和混合动力学。我们的方法基于计算A和B的值（请参见定理3.1），该值分别是系统的所有floquet乘数（特征乘数）的总和和产物。我们通过变异方法和近似理论的方法获得A（参见定理4.1）的明确表达，并通过Liouville公式对B的明确表达。此外，计算机程序旨在实现我们的算法。具体来说，只要您输入与方程参数相关的程序代码，您可以确定二阶周期性线性系统的稳定性，无论它们是离散，连续还是混合的。实际上，很少有文献处理算法来计算Floquet乘数，而是不提到为其计算机实现设计程序。我们的算法给出了所有floquet乘数的明确表达式，我们的计算机程序基于这些显式表达式的近似值。特别是，在任意离散的定期时间尺度上，我们可以进行有限数量的计算以获取floquet乘数的显式值（请参见Theorem 4.2）。因此，对于任何离散的周期系统，即使没有计算机，我们也可以准确地确定系统的稳定性！最后，在第6节中，提出了几个示例，以说明我们算法的有效性。
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.
