  Differentially positive systems are systems whose linearization along
trajectories is positive. Under mild assumptions, their solutions
asymptotically converge to a one-dimensional attractor, which must be a limit
cycle in the absence of fixed points in the limit set. In this paper, we
investigate the general connections between the (geometric) properties of
differentially positive systems and the (spectral) properties of the Koopman
operator. In particular, we obtain converse results for differential
positivity, showing for instance that any hyperbolic limit cycle is
differentially positive in its basin of attraction. We also provide the
construction of a contracting cone field.
