线性正系统的一般化及其在非线性系统中的应用：不变集与庞加莱-本迪克松性质
A Generalization of Linear Positive Systems with Applications to Nonlinear Systems: Invariant Sets and the Poincaré-Bendixson Property
AI总结：
本文提出k-正线性系统，用于分析非线性动力系统，并展示了其在不变集和庞加莱-本迪克松性质方面的应用。
摘要：
线性正系统的动力学将正正交补集映射到自身。换句话说，它将一组符号变化为零的向量集映射到自身。由此提出问题：哪些线性系统将具有k个符号变化的向量集映射到自身？本文利用合作动力系统理论和完全正矩阵理论的工具来解决这个问题，从而得到一种称为k-正线性系统的一般化正线性系统，当k=1时退化为正系统。本文描述了这种新系统在非线性动力系统分析中的应用。特别是，本文证明了此类系统具有某些显式不变集，并且对于k=2的情况，任何有界轨迹都具有庞加莱-本迪克松性质。
英文摘要：
The dynamics of linear positive systems map the positive orthant to itself. In other words, it maps a set of vectors with zero sign variations to itself. This raises the following question: what linear systems map the set of vectors with $k$ sign variations to itself? We address this question using tools from the theory of cooperative dynamical systems and the theory of totally positive matrices. This yields a generalization of positive linear systems called $k$-positive linear systems, that reduces to positive systems for $k=1$. We describe applications of this new type of systems to the analysis of nonlinear dynamical systems. In particular, we show that such systems admit certain explicit invariant sets, and for the case $k=2$ establish the Poincaré-Bendixson property for any bounded trajectory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
