  Two complementary analyses of a cyclic negative feedback system with delay
are considered in this paper. The first analysis applies the work by Sontag,
Angeli, Enciso and others regarding monotone control systems under negative
feedback, and it implies the global attractiveness towards an equilibrium for
arbitrary delays. The second one concerns the existence of a Hopf bifurcation
on the delay parameter, and it implies the existence of nonconstant periodic
solutions for special delay values. A key idea is the use of the Schwarzian
derivative, and its application for the study of Michaelis-Menten
nonlinearities. The positive feedback case is also addressed.
