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Synchronization among rhythmic elements is modeled by coupledphase-oscillators each of which has the so-called natural frequency. Asymmetric natural frequency distribution induces a continuous or discontinuoussynchronization transition from the nonsynchronized state, for instance. It hasbeen numerically reported that asymmetry in the natural frequency distributionbrings new types of bifurcation diagram having, in the order parameter,oscillation or a discontinuous jump which emerges from a partially synchronizedstate. We propose a theoretical classification method of five types ofbifurcation diagrams including the new ones, paying attention to the generalityof the theory. The oscillation and the jump from partially synchronized statesare discussed respectively by the linear analysis around the nonsynchronizedstate and by extending the amplitude equation up to the third leading term. Thetheoretical classification is examined by comparing with numerically obtainedone.

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