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

Patterns of synchronization in the hydrodynamic coupling of active colloids