Synchronization of self-sustained oscillators inertially coupled through common damped system

► We study dynamics of two indirectly coupled self-oscillating systems. ► We show the existence of both mono-stable and multi-stable synchronous regimes. ► The mechanisms of transition between various synchronous regimes are explained. ► Stability domains for in-phase and anti-phase oscillations are analytically obtained. ► The possibility of a sudden amplification of the common load amplitude is revealed.