  A system of active colloidal particles driven by harmonic potentials to
oscillate about the vertices of a regular polygon, with hydrodynamic coupling
between all particles, is described by a piece-wise linear model which exhibits
various patterns of synchronization. Analytical solutions are obtained for this
class of dynamical systems. Depending only on the number of particles, the
synchronization occurs into states in which nearest neighbors oscillate either
in-phase, or anti-phase, or in phase-locked (time-shifted) trajectories.
