# Novel transition and Bellerophon state in coupled Stuart-Landau oscillators

## TL;DR

This paper investigates synchronization transitions in coupled Stuart-Landau oscillators with different frequency distributions, revealing novel Bellerophon states with two stages of phase synchronization and the impact of distribution width on transition type.

## Contribution

It uncovers the existence of Bellerophon states with two stages in amplitude-coupled oscillators and characterizes the transition behaviors based on frequency distribution width.

## Key findings

- First-order transition for compact distributions
- Continuous transition for wide distributions
- Bellerophon state with chaotic and periodic phase synchronization stages

## Abstract

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.