  We study two models of connected pendulum clocks synchronizing their
oscillations, a phenomenon originally observed by Huygens. The oscillation
angles are assumed to be small so that the pendulums are modeled by harmonic
oscillators, clock escapements are modeled by the van der Pol terms. The mass
ratio of the pendulum bobs to their casings is taken as a small parameter.
Analytic conditions for existence and stability of synchronization regimes, and
analytic expressions for their stable amplitudes and period corrections are
derived using the Poincare theorem on existence of periodic solutions in
autonomous quasi-linear systems. The anti-phase regime always exists and is
stable under variation of the system parameters. The in-phase regime may exist
and be stable, exist and be unstable, or not exist at all depending on
parameter values. As the damping in the frame connecting the clocks is
increased the in-phase stable amplitude and period are decreasing until the
regime first destabilizes and then disappears. The results are most complete
for the traditional three degrees of freedom model, where the clock casings and
the frame are consolidated into a single mass.
