Synchronization and Chaos in Coupled Systems: The model of two coupled Brusselators

This thesis discusses problems relating to the study of coupled systems. Our research focuses on answering a specific question: Is it possible to create chaos by means of coupling dynamics, where the dynamics and the mechanisms of coupling are chosen to be as simple as possible? One of the main contributions of this thesis is to provide an affirmative answer. To be precise, we have proved theoretical results that allow us to show that in a model consisting of two Brusseltators coupled by linear diffusion there exist nilpotent singularities that explain the genesis of chaotic dynamics. We also include an exhaustive study of the singularities that arise in that model. In particular, we pay special attention to the Hopf-pitchfork singularities, contributing to the general knowledge of their bifurcation diagrams. Among the different types of Hopf-pitchfork singularities, some allow us to understand the synchronization phenomena and others become possible organizing centers of chaotic dynamics.