  In this paper, we investigate geometric properties of monotone systems by
studying their isostables and basins of attraction. Isostables are boundaries
of specific forward-invariant sets defined by the so-called Koopman operator,
which provides a linear infinite-dimensional description of a nonlinear system.
First, we study the spectral properties of the Koopman operator and the
associated semigroup in the context of monotone systems. Our results generalize
the celebrated Perron-Frobenius theorem to the nonlinear case and allow us to
derive geometric properties of isostables and basins of attraction.
Additionally, we show that under certain conditions we can characterize the
bounds on the basins of attraction under parametric uncertainty in the vector
field. We discuss computational approaches to estimate isostables and basins of
attraction and illustrate the results on two and four state monotone systems.
