Su un modello a due strati di oscillatori di Kuramoto debolmente accoppiati: Analisi e teoria della dissipazione

This thesis was written under a rather curious analytical eye on the behavior of Kuramoto oscillators. Note that ours are not the first eyes that fall on this type of oscillators, the good thing is that the dynamic system has multiple forms and questions, and, like the universe, mathematical theories continue to expand and give us more opportunities to analyze a problem in different ways, allowing us to cover every question we have. This is how we arrive at this thesis, where we work with two-layer oscillators, analyze their behavior and exploit their structure through the lens of dissipation theory. In the first part of the thesis, we present the theory surrounding the Kuramoto model, from the first questions raised about the synchronization phenomenon, to the most significant properties of the model. We also establish the mathematical framework that will be used in the rest of the chapters of the thesis. The second part of the thesis is focused on the proposed two-layer models, the work of systems over networks is quite popular nowadays since it allows us to apply them to a great variety of phenomena, such as human interactions, neuronal, electrical networks, among others. In general, they are studies focused on the general behavior of the system and not on the behavior of each layer, our interest goes beyond this uniform coupling and falls on the ability of each layer to achieve a synchronization state under a fairly weak coupling between layers, the study of the models under this perspective allows us to exploit the characteristics of its own topology. Thus, we analyze the stationary state of the system with identical oscillators and the behavior of the corresponding order parameters both analytically and via numerical simulations. Then, we study the case for non-identical oscillators, analyze the diameter of each layer and present the sufficient conditions to control the diameter. Moreover, our numerical results allow us to validate the presented theoretical derivation. Finally, in the third part of the thesis, our focus is redirected to the study of dissipation theory. We make a review of the Dissipation Function and cast the Kuramoto model in the framework of the exact response theory. We analyze the behavior of the Dissipation Function in the simplest case of two interactive oscillators as well in the presence of the $N$ coupled oscillators. Then, we compare the predictions of linear response theory with those of the exact response theory. Finally, we extend our analysis to a model constituted by two sets (or "layers") of Kuramoto oscillators, interacting with one another via a small coupling constant $\eps$. Our theoretical derivation is also complemented by numerical simulations of the Kuramoto dynamics.