A Generalization of Linear Positive Systems

The dynamics of linear positive systems maps the positive orthant to itself. Namely, it maps a set of vectors with zero sign variations to itself. Hence, a natural question is: what linear systems map the set of vectors with k sign variations to itself? To address this question we use tools from the theory of cooperative dynamical systems and the theory of totally positive matrices. Our approach yields a generalization of positive linear systems called k-positive linear systems, which reduces to positive systems for k = 1. We show an application of this new class of systems to the analysis of invariant sets in nonlinear time-varying dynamical systems.