  In this paper the existence and unicity of a stable periodic orbit is proven,
for a class of piecewise affine differential equations in dimension 3 or more,
provided their interaction structure is a negative feedback loop. It is also
shown that the same systems converge toward a unique stable equilibrium point
in dimension 2. This extends a theorem of Snoussi, which showed the existence
of these orbits only. The considered class of equations is usually studied as a
model of gene regulatory networks. It is not assumed that all decay rates are
identical, which is biologically irrelevant, but has been done in the vast
majority of previous studies. Our work relies on classical results about fixed
points of monotone, concave operators acting on positive variables. Moreover,
the used techniques are very likely to apply in more general contexts, opening
directions for future work.
