title: Inducing chaos in a gene regulatory network by coupling an oscillating dynamics with a hysteresis-type one.
abstract: In this paper, we investigate the chaotic behavior of a gene regulatory network modeled by four differential equations and seventeen parameters. This network, called [Formula: see text]-system, has been designed to couple in a simple way an oscillating system with one having a bistable switch. After having studied it analytically, we exhibit (by a constructive proof) the mechanism responsible of chaos for a general differential system presenting such a coupling. Namely, given a generic one-parameter family of smooth vector fields on [Formula: see text] presenting a Hopf bifurcation, we prove that under an assumption on the Jacobian at the bifurcation point, we can create such a chaotic system by perturbing the parameter thanks to a hysteresis-type dynamics. Finally, we numerically show that the mechanism highlighted previously takes place in the [Formula: see text]-system, for a particular set of values of its parameters.
authors.affiliation: Mathematics Laboratory J.A Dieudonné, University of Nice Sophia Antipolis, UMR CNRS 7351, 06108 , Nice Cedex 02, France, Camille.POIGNARD@unice.fr.
authors.forename: Camille
authors.initials: C
authors.lastname: Poignard
mesh_terms: D005246:Feedback; D005786:Gene Expression Regulation / Q000235:genetics*; D053263:Gene Regulatory Networks / Q000235:genetics*; D008957:Models, Genetic; D017711:Nonlinear Dynamics