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

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