A generalization of linear positive systems with applications to nonlinear systems: Invariant sets and the Poincaré–Bendixson property

The dynamics of linear positive systems map the orthant to itself. In other words, it maps a set vectors with zero sign variations This raises following question: what $k$ itself? We address this question using tools from theory cooperative dynamical and totally matrices. yields generalization called $k$-positive systems, that reduces for $k=1$. describe applications new type analysis nonlinear systems. particular, we show such admit certain explicit invariant sets, case $k=2$ establish Poincar\'{e}-Bendixson property any bounded trajectory.