Entangled dynamical systems on networks: a density matrix approach

Synchronization of an ensemble of oscillators is an emergent phenomenon present in several fields, ranging from social and physical to biological and technological systems. The most successful approach to describe how coherent behavior emerges in these complex systems is given by the paradigmatic Kuramoto model. Despite in the last years it has seen an increasing interest growing around it and a lot of different modifications to the original model have been formulated, there is still much to uncover. Particularly, the case in which two different systems, modeled as complex networks, are subjected to an external field has not received much attention. Here, we go beyond its basic interpretation by including modifications, which involve the insertion of non-trivial correlations between different units, that make the model more suitable to describe a lot of real world and widespread phenomena, such as shared experience, birds flocking and insects swarming. By combining standard analysis techniques with the new powerful framework of the density matrix we aim to describe the emergence of collective behaviors arising from the synergy of both local and global scales together with thermodynamic properties. We show that our formulation unravels rich dynamical features with also the presence of complex and chaotic behaviors.