  We introduce the notion of non-oscillation, propose a constructive method for
its robust verification, and study its application to biological interaction
networks (also known as, chemical reaction networks). We begin by revisiting
Muldowney's result on non-existence of periodic solutions based on the study of
the variational system of the second additive compound of the Jacobian of a
nonlinear system. We show that exponential stability of the latter rules out
limit cycles, quasi-periodic solutions, and broad classes of oscillatory
behavior. We focus then on nonlinear equations arising in biological
interaction networks with general kinetics, and we show that the dynamics of
the aforementioned variational system can be embedded in a linear differential
inclusion. We then propose algorithms for constructing piecewise linear
Lyapunov functions to certify global robust non-oscillatory behavior. Finally,
we apply our techniques to study several regulated enzymatic cycles where
available methods are not able to provide any information about their
qualitative global behavior.
