This talk will be divided in two parts, respectively on the desynchronization and synchronization of systems. The entire talk will last around 2 hours.
1) Inducing chaos in a gene regulatory network by coupling an oscillating dynamics with a hysteresis-type one. In this first part we deal (under the biological viewpoint) with the desynchronization of stable systems, which consists in inducing chaos in them. We study this problem on a gene regulatory network called V-system, modeled by a differential system with 4 equations and 17 parameters, invented in order to couple in a very simple way, a Hopf bifurcation and a Hysteresis-type dynamics. After having proved that a vector field on R^n admitting such a coupling may (under some generic conditions) presents a Horseshoe-type dynamics, we give a set of parameters for which the associated V-system satisfies these conditions and show numerically that the mechanism responsible of the chaotic motion occurs in this system.
2)Discrete synchronization of hierarchically organized dynamical systems. In this second part, i will present the synchronization of an infinite number of systems in the discrete setting. We first define a hierarchical structure for a set of 2^n systems by a matrix representing the steps of a matching process in groups of size two. This leads us naturally to the case of a Cantor set of systems, for which we obtain a global synchronization result generalizing the finite case. Then, we deal with the situation where some defects appear in the hierarchy, that is to say when some links between certain systems are broken. We prove we can afford an infinite number of such broken links while keeping a local synchronization, providing they are only present at the first N stages of the hierarchy (for a fixed integer N) and they are enough spaced out in these stages.