Synchronization in Small, Fully Coupled Systems of Discrete-Time Kuramoto Oscillators

Emergent behavior of many networked entities can be modeled using a network of phase coupled oscillators, for example, the Kuramoto oscillator. The discrete-time version of the Kuramoto model is investigated here through iterated maps. The existence of a constant of motion is used to simplify the analysis. The fixed points and their stability are determined analytically. To provide a measure of synchronization, the order parameter is calculated at the fixed points. The order parameter provides a natural classification of the fixed points into balanced, unbalanced and synchronized sets. The periodic orbits and chaotic behavior of the system are also studied. Importantly, we show that by describing the symmetries of the three-oscillator model, the toroidal phase space can be fully tiled with the so-called fundamental set (a trapezoid) and its affine transformations.