  We consider the macroscopic regimes and the scenarios for the onset and the
suppression of collective oscillations in a heterogeneous population of active
rotators, comprised of excitable or oscillatory elements. We analyze the system
in the continuum limit within the framework of Ott-Antonsen reduction method,
determining the states with a constant mean field and their stability
boundaries in terms of the characteristics of the rotators' frequency
distribution. The system is established to display three macroscopic regimes,
namely the homogeneous stationary state, the oscillatory state and the
heterogeneous stationary state, whereby the transitions between the
characteristic domains involve a complex bifurcation structure, organized
around three codimension-two bifurcation points: a Bogdanov-Takens point, a
cusp point and a fold-homoclinic point. Apart from the monostable domains, our
study also reveals two domains admitting bistable behavior, manifested as
coexistence between the two stationary solutions, or between a stationary and a
periodic solution. It is shown that the collective mode may emerge via two
generic scenarios, guided by a SNIPER or the Hopf bifurcation, such that the
transition from the homogeneous to the heterogeneous stationary state under
increasing diversity may follow the classical paradigm, but may also be
hysteretic. We demonstrate that the basic bifurcation structure holds
qualitatively in presence of small noise or small coupling delay, with the
boundaries of the characteristic domains shifted compared to the noiseless and
delay-free case.
