Winfree model

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 S and I are the sensitivity and influence functions, respectively. In the literatures, S and I 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:

 \theta_i \to \theta_i + c \quad \mbox{for all} \quad i=1, \cdots, N.


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

 \sum_{i=1}^N \dot \theta_i = \sum_{i=1}^N \Omega_i.


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 \rho:

 \rho_i := \lim_{t\to\infty} \frac{\theta_i (t)}{t}


Let \mathcal N = \{1, \cdots, N\} be the set of oscillators. We classify the following four types of synchronization:
(1) Complete Oscillator Death State:

 |\{ i \in \mathcal N : \rho_i = 0 \} | = N


(2) Partial Oscillator Death State:

 2\leq |\{ i \in \mathcal N : \rho_i = 0 \} | < N


(3) Complete Phase-locked State:

 | \{ i \in \mathcal N : \rho_i = \rho \} | = N \quad \mbox{for} \quad \rho\neq 0.


(4) Partial Phase-locked State:

 2\leq | \{ i \in \mathcal N : \rho_i = \rho \} | < N \quad \mbox{for} \quad \rho\neq 0.

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