Stability of geometric flows on the circle

In this paper we prove a general stability result for higher order geometric flows on the circle, which basically states that if the initial condition is close to a round circle, the curve evolves smoothly and exponentially fast towards a circle, and we improve on known convergence rates (which we believe are almost sharp). The polyharmonic flow is an instance of the flows to which our result can be applied (and of course we present some other examples).