Patterns of synchronization in the hydrodynamic coupling of active   colloids
  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. 
Giovanni M. Cicuta and Enrico Onofri and Marco Cosentino Lagomarsino   and Pietro Cicuta
cond-mat.stat-mech cond-mat.soft math-ph math.MP