An approach to normal forms of Kuramoto model with distributed delays and the effect of minimal delay

Heterogeneous delays with positive lower bound (gap) are taken into consideration in Kuramoto oscillators. We first establish a perturbation technique, by which universal normal forms and detailed dynamical behavior of this model can be obtained easily. Theoretically, a hysteresis loop is found near the subcritically bifurcated coherent state on the Ott-Antonsen's manifold. For Gamma distributed delay with fixed variance and mean, we find large gap destroys the loop and significantly increases in the number of coexisted coherent attractors. This result is also explained in the viewpoint of excess kurtosis.