Coupled Systems and Morphogenesis Self-Organization in Biological Systems

In this dissertation, we investigate a class of perturbed interconnected mean-field systems, commonly referred to as coupled systems. Under suitable assumptions, we establish the existence of an invariant open set under the flow of the perturbed system. In other words, we prove that the distance between the components of an orbit remains uniformly bounded, a property known as synchronization. The main tool used is the perturbation method. Notably, the synchronization result does not hold trivially in the unperturbed system. We also apply a fixed point theorem to demonstrate the existence of periodic orbits on the torus. Furthermore, we analyze both stability and exponential stability of such systems by studying the dynamics of associated linear systems.