Inducing chaos in a gene regulatory network by coupling an oscillating dynamics with a hysteresis-type one
In this paper, we investigate the chaotic behavior of a gene regulatory network modeled by four differential equations and seventeen parameters. This network, called $$\mathrm{V}$$V-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 $$\mathbb R ^n$$Rn 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 $$\mathrm{V}$$V-system, for a particular set of values of its parameters.