We show that active, self-propelled particles that are connected together to
form a single chain that is anchored at one end can produce the graceful
beating motions of flagella. Changing the boundary condition from a
clamp to a pivot at the anchor leads to steadily rotating tight coils. Strong
noise in the system disrupts the regularity of the oscillations. We use a combination of detailed numerical simulations, mean-field scaling analysis and
first passage time theory to characterize the phase diagram as a function
of the filament length, passive elasticity, propulsion force and noise. Our
study suggests minimal experimental tests for the onset of oscillations in
an active polar chain.