  When modeling the classical Kuramoto model, one of the key features is the
tendency to synchronize. Accordingly, the most well-adopted choice of the
coupling function is the sine function. Due to the oddness of the sine
function, the synchronized frequency would be the average of all the natural
frequencies. In this article, we study the synchronization behaviors of the
Kuramoto model with a pure competition coupling function. Namely, instead of
the sine function, we choose $\max \{0, \sin \theta \}$ to be the coupling
function. This indicates the relation of pure competition between oscillators.
We prove asymptotical phase synchronization for identical oscillators and
asymptotical frequency synchronization for non-identical oscillators under
reasonable sufficient conditions. In particular, under our sufficient
conditions, the synchronized frequency is the maximal frequency of all the
natural frequencies. On the other hand, in the parameter regime which is out of
the scope of the analysis of our theorems, it is possible that the synchronized
frequency could be larger than the maximal frequency of the natural frequencies
of all the oscillators. In this article, we also provide numerical experiments
to support the analysis of our theorem and to demonstrate the aforementioned
phenomenon.
