  In this paper, we consider the systems with trajectories originating in the
nonnegative orthant becoming nonnegative after some finite time transient.
First we consider dynamical systems (i.e., fully observable systems with no
inputs), which we call eventually positive. We compute forward-invariant cones
and Lyapunov functions for these systems. We then extend the notion of
eventually positive systems to the input-output system case. Our extension is
performed in such a manner, that some valuable properties of classical
internally positive input-output systems are preserved. For example, their
induced norms can be computed using linear programming and the energy functions
have nonnegative derivatives.
