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We consider the macroscopic regimes and the scenarios for the onset and thesuppression of collective oscillations in a heterogeneous population of activerotators, comprised of excitable or oscillatory elements. We analyze the systemin the continuum limit within the framework of Ott-Antonsen reduction method,determining the states with a constant mean field and their stabilityboundaries in terms of the characteristics of the rotators' frequencydistribution. The system is established to display three macroscopic regimes,namely the homogeneous stationary state, the oscillatory state and theheterogeneous stationary state, whereby the transitions between thecharacteristic domains involve a complex bifurcation structure, organizedaround three codimension-two bifurcation points: a Bogdanov-Takens point, acusp point and a fold-homoclinic point. Apart from the monostable domains, ourstudy also reveals two domains admitting bistable behavior, manifested ascoexistence between the two stationary solutions, or between a stationary and aperiodic solution. It is shown that the collective mode may emerge via twogeneric scenarios, guided by a SNIPER or the Hopf bifurcation, such that thetransition from the homogeneous to the heterogeneous stationary state underincreasing diversity may follow the classical paradigm, but may also behysteretic. We demonstrate that the basic bifurcation structure holdsqualitatively in presence of small noise or small coupling delay, with theboundaries of the characteristic domains shifted compared to the noiseless anddelay-free case.

Two Scenarios for the Onset and Suppression of Collective Oscillations in Heterogeneous Populations of Active Rotators