Phase diagram of a generalized Winfree model

We study the phase diagram of a generalized Winfree model. The modification is such that the coupling depends on the fraction of synchronized oscillators, a situation which has been noted in some experiments on coupled Josephson junctions and mechanical systems. We let the global coupling $k$ be a function of the Kuramoto order parameter $r$ through an exponent $z$ such that $z=1$ corresponds to the standard Winfree model, $z<1$ strengthens the coupling at low $r$ (low amount of synchronization), and at $z>1$, the coupling is weakened for low $r$. Using both analytical and numerical approaches, we find that $z$ controls the size of the incoherent phase region and that one may make the incoherent behavior less typical by choosing $z<1$. We also find that the original Winfree model is a rather special case; indeed, the partial locked behavior disappears for $z>1$. At fixed $k$ and varying $\gamma$, the stability boundary of the locked phase corresponds to a transition that is continuous for $z<1$ and first order for $z>1$. This change in the nature of the transition is in accordance with a previous study of a similarly modified Kuramoto model.

Figure 1

The limiting value ${\gamma }_d$ as a function of $z$. The Winfree model is obtained for $z=1$.

Figure 2

Phase diagram of the generalized Winfree model. Results are for $z=0.5$ (top left), $z=1$ (top right), $z=1.5$ (bottom left), and $z=2$ (bottom right). $\mathrm{Dt}=\text{death}$ phase, $\mathrm{PD}=\text{partial}$ death, $\mathrm{In}=\text{incoherence}$, $\mathrm{Lk}=\text{locked}$ phase, and $\mathrm{PL}=\text{partial}$ locking. The phase diagram at $z=1$ coincides with that depicted in [8]. In the case $z=1.5$ the stability boundary between partial locking and incoherence has been drawn as the dash-dotted line because it is not observed in simulations (for $z>1$ the partial locking region is absent; see the text).

Figure 3

The order parameter (time-averaged) $r$ versus $\gamma$, for low $\gamma$ and fixed $k=0.4$, is reported for $z=0.5$ (top left, continuous transition between the locked phase and partial locked phase), $z=1$ (top right), $z=1.5$ (bottom left), and $z=2$ (bottom right). In the last three cases there is a first-order transition between the locked phase and incoherence phase.