Peculiarities of the transitions to synchronization in coupled systems with amplitude death.

The paper presents the results of the study of the sequences of bifurcation leading to the synchronization and amplitude death in a system of two dissipatively coupled self-sustained oscillators with inertial nonlinearity. Two types of synchronizations tongues have been identified. In one of them phase locking regions exist where the synchronization is achieved by the saddle-node bifurcation and regions where the transition to synchronization leads through Neimark-Sacker bifurcation. In the second type of the tongues there are only phase locking regions. It has been shown that for a weak non-identity of the system parameters, the first type tongues merge together. The transition between the synchronization tongues can occur without bifurcations, i.e., transition between the synchronized regimes with different periods of oscillations can occur gradually.