The piecewise-linear differential equations treated by the qualitative simulation method have discontinuities in their right-hand side. In collaboration with J.-L. Gouzé (INRIA Sophia-Antipolis) and T. Sari (université de Haute-Alsace, Mulhouse), the mathematical problems related to the discontinuities have been studied, using the concept of Filippov solutions. This work has been continued and extended in a joint research project, called GDyn: Dynamic analysis of genetic regulatory networks, which was funded in the framework of the Actions de Recherche Coopératives of INRIA. One of the results of GDyn has been the characterization of the stability of equilibrium points located on the planes of discontinuity of the system. In particular, we have tried to answer the question whether the stability of an equilibrium point can be inferred from topological properties of the state transition graph generated by the qualitative simulation method.

State transition graph describing the qualitative dynamics of a genetic regulatory network.

Literature

R. Casey, H. de Jong, J.-L. Gouzé (2004), Piecewise-linear models of genetic regulatory networks: Equilibria and their stability, INRIA, RR-5353.

H. de Jong, J.-L. Gouzé, C. Hernandez, M. Page, T. Sari, J. Geiselmann (2004), Qualitative simulation of genetic regulatory networks using piecewise-linear models, Bulletin of Mathematical Biology, 66(2):301-340.

J.-L. Gouzé, T. Sari (2003), A class of piecewise linear differential equations arising in biological models, Dynamical Systems, 17(4):299-316.