  A rationale is provided for the emergence of synchronization in a system of
coupled oscillators in a stick-slip motion. The single oscillator has a limit
cycle in a region of the state space for each parameter set beyond the
supercritical Hopf bifurcation. The two-oscillator system that has similar
weakly coupled oscillators exhibits synchronization in a parameter range. The
synchronization has an anti-phase nature for an identical pair. However, it
tends to be more in-phase for a non-identical pair with a rather weak coupling.
A system of three identical oscillators (1, 2, and 3) coupled in a line (with
two springs k12=k23) exhibits synchronization with two of them (1 and 2 or 2
and 3) being nearly in-phase. These collective behaviours are systematically
estimated using the phase reduction method.
