Chaotic Attractor in the Kuramoto Model

The Kuramoto model of globally coupled phase oscillators is an essentially nonlinear dynamical system with a rich dynamics including synchronization and chaos.We study the Kuramoto model from the standpoint of bifurcation and chaos theory of low-dimensional dynamical systems. We find a chaotic attractor in the four-dimensional Kuramoto model and study its origin. The torus destruction scenario is one of the major mechanisms by which chaos arises. L. P. Shilnikov has made decisive contributions to its discovery. We show also that in the Kuramoto model the transition to chaos is in accordance with the torus destruction scenario. We present the general bifurcation diagram containing phase chaos, Cherry flow as well as periodic and quasiperiodic dynamics.