  It has been shown that self-assembled chains of active colloidal particles
can present sustained oscillations. These oscillations are possible because of
the effective diffusiophoretic forces that mediate the interactions of colloids
do not respect the action--reaction principle and hence, a Hopf bifurcation is
possible even for overdamped dynamics. Anchoring the particles in one extreme
breaks the head-tail symmetry and the oscillation is transformed into a
traveling wave pattern and thus the chain behaves like a beating cilium. The
net force on the anchor, estimated using the resistive force theory, vanishes
before the bifurcation and thereafter grows linearly with the bifurcation
parameter. If the mobilities of the particles on one extreme are reduced to
mimic an elongated cargo, the traveling wave generates a net velocity on the
chain that now behaves like a moving flagellum. The average velocity again
grows linearly with the bifurcation parameter. Our results demonstrate that
simplified systems, consisting only of a few particles with non-reciprocal
interaction and head-tail asymmetry show beating motion and self-propulsion.
Both properties are present in many non-equilibrium models thus making our
results a general feature of active matter.
