  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.
