A Generalization of Linear Positive Systems with Applications to Nonlinear Systems: Invariant Sets and the Poincaré-Bendixson Property
Provides a new theoretical framework for analyzing nonlinear dynamical systems, particularly those exhibiting oscillatory behavior, by extending the theory of positive systems.
The paper generalizes linear positive systems to k-positive linear systems, which map sets of vectors with k sign variations to themselves. For k=2, they prove the Poincaré-Bendixson property for bounded trajectories, enabling analysis of nonlinear systems.
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.