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

A robust Lyapunov criterion for non-oscillatory behaviors in biological interaction networks