A Generalization of Linear Positive Systems with Applications to Nonlinear Systems: Invariant Sets and the Poincar\'{e}-Bendixson Property

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 Poincar\'{e}-Bendixson property for any bounded trajectory.