Analytical characterization of the simplicial Kuramoto model

Synchronization is a ubiquitous phenomenon, universally found both in natural and human-engineered systems: from firing neurons to the twinkling of fireflies, from metronomes to power grids, from applauding audiences to the circadian rhythms of plants and animals. Patterns and order magically emerge from the disordered interplay of many interacting parts, with a complexity which seems to give no hope for simple models. The Kuramoto model is one of the hallmarks of complex systems’ theory, with its simplicity, richness of behavior and surprising modeling power. It allows us to describe the onset of synchronization in systems of oscillators which interact in pairs according to a determined network topology. A particular reformulation of the Kuramoto model naturally lends itself to a generalization which, on the recent wave of research on higher-order systems, allows one to go beyond pairwise interactions and consider oscillators influencing each other in groups. Such higher-order interactions can be easily described with simplicial complexes, discrete combinatorial objects which generalize graphs, from nodes and edges, to triangles, tetrahedra and so on. The rich theory of discrete exterior calculus, which brings classical calculus on manifolds to the discrete domains of simplicial complexes, provides us with a vast set of tools and ideas which can help us to study the simplicial Kuramoto model. Interesting relations between the topology of the complex and the synchronization properties of the model emerge. In the thesis, after having introduced Hodge decomposition and discrete exterior calculus, we explore the properties of the simplicial Kuramoto model on weighted simplicial complexes, with a particular focus on its equilibrium properties. We study the peculiar character of simplicial interactions, generalize known results to find new notions of synchronization and phase-locking, together with necessary and sufficient conditions for their onset. We investigate the equilibrium phase transition which occurs when the strength of the interaction is increased, and its effect on the order parameter. Finally, we consider the case where an external frustration influences the interactions, study how it reshapes the equilibrium configurations of the system and how it can be used in a synchronization control perspective.