The polyharmonic Kuramoto model

The Kuramoto model is the most well-established framework for describing the emergence of synchronization in large populations of coupled oscillators. Among its many generalizations, the introduction of higher-harmonic interactions extends its ability to capture a broader range of collective behaviors. This thesis investigates the role of higher harmonics in synchronization phenomena, first by analyzing the effect of individual harmonic contributions to the coupling function and subsequently by deriving several properties of multi-harmonic extensions, with particular emphasis on the Biharmonic model, i.e. the Kuramoto model augmented by a second-harmonic coupling term. The investigation has been carried out using standard techniques from dynamical systems, such as stability analysis and bifurcation theory, along with statistical-mechanics approaches, chiefly self-consistency. Numerical simulations based on Euler's algorithm have been employed to validate analytical predictions. The study clarifies the origin of clustering in single-harmonic models and proves their equivalence to the classical Kuramoto model. In the thermodynamic limit, the Biharmonic model is shown to display a discontinuous phase transition within a non trivial region of the coupling space. Furthermore, results concerning multi-harmonic models have been achieved, providing the groundwork for future investigations of higher-order synchronization phenomena.