  Synchronization in a population of oscillators with hyperbolic chaotic phases
is studied for two models. One is based on the Kuramoto dynamics of the phase
oscillators and on the Bernoulli map applied to these phases. This system
possesses an Ott-Antonsen invariant manifold, allowing for a derivation of a
map for the evolution of the complex order parameter. Beyond a critical
coupling strength, this model demonstrates bistability synchrony-disorder.
Another model is based on the coupled autonomous oscillators with hyperbolic
chaotic strange attractors of Smale-Williams type. Here a disordered
asynchronous state at small coupling strengths, and a completely synchronous
state at large couplings are observed. Intermediate regimes are characterized
by different levels of complexity of the global order parameter dynamics.
