In 1967, A. Winfree proposed a synchronization model, which is known for the first mathematical model describing synchronous phenomena.

\begin{equation}\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,

\end{equation}

where and are the sensitivity and influence functions, respectively. In the literatures, and are often assumed to be

\begin{equation}\label{winf-2}

S(\theta) = -\sin\theta, \quad I(\theta) = 1+\cos\theta.

\end{equation}

Unlike the Kuramoto model, the Winfree model hasn't had big attentions so far. The Winfree model doesn't have a good properties that the Kuramoto model has. For example, Since the dynamics of the Kuramoto model depend on the difference of the phases, it has an invariance of translation:

Moreover, the oddness of sine function yields the conservation of total phase velocity:

Such good properties doesn't exists in the Winfree model and it makes difficulty in analysis.

However, the Winfree model has much richer and various appearance in synchronization. To classify the synchronous phenomena in the Winfree model, we define the rotation number :

Let be the set of oscillators. We classify the following four types of synchronization:

(1) Complete Oscillator Death State:

(2) Partial Oscillator Death State:

(3) Complete Phase-locked State:

(4) Partial Phase-locked State:

Depend on the magnitude of the coupling strength and the distribution of the natural frequencies , the Winfree model shows different appearances.

Pingback: Emergence of phase-locked states for the Winfree model in a large coupling regime | 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Emergent dynamics of Winfree oscillators on locally coupled networks | 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Collective synchronization of classical and quantum oscillators | 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Emergence of partial locking states from the ensemble of Winfree oscillators | 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Emergence of phase-locked states for the Winfree model in a large coupling regime | 박진영(Jinyeong Park, 朴 鎭永)