A class of piecewise linear differential equations arising in biological models

We investigate the properties of the solutions of a class of piecewise-linear differential equations. The equations are appropriate to model biological systems (e.g. genetic networks) in which there are switch-like interactions between the elements. The analysis uses the concept of Filippov solutions of differential equations with a discontinuous right-hand side. It gives an insight into the so-called singular solutions which lie on the surfaces of discontinuity. We show that this notion clarifies the study of several examples studied in the literature.