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

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, or oscillation, from the nonsynchronized state, for example. It has been numerically reported that asymmetry in the natural frequency distribution introduces new types of bifurcation diagrams with, in the order parameter, oscillation or a discontinuous jump which emerges from a partially synchronized state. We propose a theoretical classification method for 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 through comparison with the numerically obtained classification