# Classification of bifurcation diagrams in coupled phase-oscillator models with asymmetric natural frequency distributions

## TL;DR

This paper develops a theoretical framework to classify five types of bifurcation diagrams in coupled phase-oscillator models with asymmetric natural frequency distributions, expanding understanding of synchronization phenomena.

## Contribution

It introduces a general classification method for bifurcation diagrams, including new types caused by asymmetry, using linear analysis and amplitude equations.

## Key findings

- Five bifurcation diagram types identified
- Theoretical classification matches numerical results
- New bifurcation behaviors explained by asymmetry

## Abstract

Synchronization among rhythmic elements is modeled by coupled phase-oscillators each of which has the so-called natural frequency. A symmetric natural frequency distribution induces a continuous or discontinuous synchronization transition from the nonsynchronized state, for instance. It has been numerically reported that asymmetry in the natural frequency distribution brings new types of bifurcation diagram having, in the order parameter, oscillation or a discontinuous jump which emerges from a partially synchronized state. We propose a theoretical classification method of five types of bifurcation diagrams including the new ones, paying attention to the generality of the theory. The oscillation and the jump from partially synchronized states are discussed respectively by the linear analysis around the nonsynchronized state and by extending the amplitude equation up to the third leading term. The theoretical classification is examined by comparing with numerically obtained one.