3.4. Dynamics of Kuramoto

There are a number of alternative cases to consider while investigating the dynamics of the Kuramoto Model. We can say the following, for small values of N [7];

• K > K_c always leads to synchronised behaviour (either partial synchrony, or total synchrony).
  
• K = 0 removes coupling from the system, reducing behaviour to that of simple uncoupled oscillators.
  
• 0 < K < K_c with N < 4 will lead to periodic anti-synchoronous behaviour.
  
• 0 < K < K_cr < K_c with N > 3 often (but not always) encourages a variety of chaotic behaviours.
  
• K_cr < K < K_c with N > 3 leads to a stable periodic orbit.
  

When the coupling constant K > K_c the dynamics of the model are relatively straight-forward. Regardless of the number of oscillators in the system, the a number of the oscillators will synchronise and remain synchronised as t → ∞. The higher we make K, the more oscillators will be synchronised until K > K_max, when we reach total synchrony.

Considering K < K_c gives rise to a variety of possible dynamics. If the coupling constant K = 0 the Kuramoto model is reduced to simple uncoupled oscillator dynamics represented by θ_i = ω_i for i=1,...,N. Hence, the system exhibits periodic or quasi periodic rotations on the N-dimensional torus T^N [2].

When K increases above zero (but remains small, ie. 0 < K < K_cr) the dynamics of the system are perturbed and for N > 3 they often become chaotic [7]. This transition to chaos occurs in accordance to the Afraimovich-Shilnikov torus breakdown scenario. When K_cr < K < K_c we always get a stable periodic orbit.