  We study synchronization in a system of Stuart-Landau oscillators with
frequency-weighted coupling. For three typical unimodal frequency
distributions, namely, the Lorentzian, the triangle, and the uniform, we found
that the first-order transition occurs when the frequency distribution is
relatively compact, while the synchronization transition is continuous when it
is relatively wide. In both cases, there is a regime of Bellerophon state
between the incoherent state and the synchronized state. Remarkably, we
revealed novel transition behavior for such coupled oscillators with
amplitudes, i.e., the regime of Bellerophon state actually contains two stages.
In the first stage, the oscillators achieve chaotic phase synchronization;
while in the second stage, oscillators form periodical phase synchronization.
Our results suggest that Bellerophon state also exists in coupled oscillators
with amplitude dynamics.
