On the Equivalence between the Sigmoidal Approach and Utkin's Approach for Piecewise-Linear Models of Gene Regulatory Networks

This paper is concerned with piecewise-linear dynamical systems modeling a simple class of gene regulatory networks. One of the main issues when dealing with these problems is that the vector field is not defined on the discontinuity hyperplanes. Two different methods are usually employed in the literature to overcome this issue: Filippov's convexification approach and the steep sigmoidal approach. A particular selection of Filippov's vector field, namely Utkin's vector field, will be of interest to us. Our purpose is twofold: show that Utkin's vector field is well defined on the intersection Σ of two discontinuity hyperplanes (under assumptions of attractivity) and prove that, for Σ nodally attractive and attractive with three surfaces, Utkin's approach and the steep sigmoidal approach are equivalent, i.e., the corresponding solutions on Σ are the same. This allows us to study the piecewise dynamical system, and hence the gene regulatory network it models, with no ambiguity.