  Dynamical systems whose linearizations along trajectories are positive in the
sense that they infinitesimally contract a smooth cone field are called
differentially positive. The property can be thought of as a generalization of
monotonicity, which is differential positivity in a linear space with respect
to a constant cone field. Differential positivity places significant
constraints on the asymptotic behavior of trajectories under mild technical
conditions. This paper studies differentially positive systems defined on Lie
groups. The geometry of a Lie group allows for the generation of invariant cone
fields over the tangent bundle given a single cone in the Lie algebra. We
outline the mathematical framework for studying differential positivity of
discrete and continuous-time dynamics on a Lie group with respect to an
invariant cone field and motivate the use of this analysis framework in
nonlinear control, and, in particular in nonlinear consensus theory. We also
introduce a generalized notion of differential positivity of a dynamical system
with respect to an extended notion of cone fields generated by cones of rank
$k$. This new property provides the basis for a generalization of differential
Perron-Frobenius theory, whereby the Perron-Frobenius vector field which shapes
the one-dimensional attractors of a differentially positive system is replaced
by a distribution of rank $k$ that results in $k$-dimensional integral
submanifold attractors instead.
