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The celebrated Kuramoto model provides an analytically tractable framework tostudy spontaneous collective synchronization and comprises globally coupledlimit-cycle oscillators interacting symmetrically with one another. TheSakaguchi-Kuramoto model is a generalization of the basic model that considersthe presence of a phase-lag parameter in the interaction, thereby making itasymmetric between oscillator pairs. Here, we consider a furthergeneralization, by adding an interaction that breaks the phase-shift symmetryof the model. The highlight of our study is the unveiling of a very richbifurcation diagram comprising of both oscillatory and non-oscillatorysynchronized states as well as an incoherent state: There are regions oftwo-state as well as an interesting and hitherto unexplored three-statecoexistence arising from asymmetric interactions in our model.

The Sakaguchi-Kuramoto model in presence of asymmetric interactions that break phase-shift symmetry