Mathematical Aspects of Collective Dynamics:
Kinetic Description and Fractional Diffusion

The synchronization problem for Kuramoto oscillators

Javier Morales

University of Maryland

Collective phenomena such as aggregation, flocking, and synchronization are ubiquitous in natural biological, chemical, and mechanical systems--e.g., the flashing of fireflies, chorusing of crickets, synchronous firing of cardiac pacemakers, and metabolic synchrony in yeast cell suspensions. The Kuramoto model introduced by Yoshiki Kuramoto is one of the first theoretical tools developed to understand such phenomena and has recently gained extensive attention in the physical and mathematical community. Moreover, it has become the starting point of several generalizations that have applications ranging from opinion dynamics to the development of human-made interacting multi-agent systems of UAVs and data clustering. In this talk, we will review the state of the art for the synchronization problem of the Kuramoto model at the kinetic and particle level. Additionally, we will introduce new developments and variational techniques for the dynamics of this model.