In 1970, Binyamin Schwarz defined and analyzed totally positive differential systems (TPDSs), i.e., linear time-varying systems whose transition matrix is totally positive. He showed that any solution of a TPDS satisfies a sign variation diminishing property with respect to the standard number of sign variations. It has been recently shown that several important results on entrainment [stability] in time-varying [time-invariant] nonlinear tridiagonal cooperative systems follow from the fact that the variational equation associated with these nonlinear systems is a TPDS. Thus, the number of sign variations in the vector of derivatives can be used as an integer-valued Lyapunov function. Here we develop the theory of linear cyclic variation diminishing differential systems (CVDDSs). These are systems whose transition matrix satisfies a variation diminishing property with respect to the cyclic number of sign variations. Thus, the cyclic number of sign variations can be used as an integer-valued Lyapunov function for any vector solution of a CVDDS. We show that several known classes of nonlinear cooperative dynamical systems have a variational equation, which is a CVDDS.

中文翻译：

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具有循环符号变分递减性质的动力系统


1970年，Binyamin Schwarz定义并分析了全正微分系统（TPDS），即转移矩阵全为正的线性时变系统。他证明了 TPDS 的任何解都满足相对于符号变化的标准数量的符号变化递减性质。最近的研究表明，时变[时不变]非线性三对角协作系统中夹带[稳定性]的几个重要结果源于与这些非线性系统相关的变分方程是TPDS这一事实。因此，导数向量中符号变化的数量可以用作整数值李亚普诺夫函数。在这里，我们发展了线性循环变分递减微分系统（CVDDS）的理论。这些系统的转移矩阵满足关于符号变化的循环数的变化递减性质。因此，符号变化的循环数可以用作 CVDDS 的任何向量解的整数值 Lyapunov 函数。我们证明了几种已知类型的非线性协作动力系统都有一个变分方程，即 CVDDS。