# Winfree model

In 1967, A. Winfree proposed a synchronization model, which is known for the first mathematical model describing synchronous phenomena.
\label{winf-1}
\dot \theta_i =\Omega_i + \frac{K}{N} S(\theta_i)\sum_{j=1}^N I(\theta_j), \quad i=1, \cdots, N,

where $S$ and $I$ are the sensitivity and influence functions, respectively. In the literatures, $S$ and $I$ are often assumed to be
\label{winf-2}
S(\theta) = -\sin\theta, \quad I(\theta) = 1+\cos\theta.

# Kuramoto model

Synchronization is collective behaviors of periodic motions, which is easily observed in many biologiclal, social and physical systems. Y. Kuramoto proposed a mathmatical model for synchronization in 1975. Kuramoto model describes the periodic motion as a rotation of phase on the unit circle. By using polar coordinate, the dynamics can be expressed in the following form:
\label{Ku-1}
\dot\theta_i = \Omega_i + \frac{K}{N} \sum_{j=1}^N \sin(\theta_j - \theta_i) \quad \mbox{for} \quad  i=1, \cdots, N

$\Omega_i$ and $K$ Kare called the natural frequency and the coupling strength, respectively.