I am trying to study shape deformations of a closed circle, which are periodic but non-reciprocal. A similar question has already been asked and answered. But I am now interested in studying the time-evolution in addition. By periodic I mean periodic in time. I.e. After one cycle, the circle should have the same shape as in the beginning. However it should also be non-reciprocal. Initially I thought one could represent such deformations by considering two Fourier Series: One in time and one depending on the angle. While such a representation is cyclic it is unfortunately also reciprocal, meaning that it is invariant under time-reversal. Do such deformations exist at all? There are a few conditions that need to be fulfilled:
- There should be no loose ends and if one plots the radius vs angle one gets a smooth function.
- The function mapping the angle to the radius is single-valued for each angle.