Stable polygons of cyclic pursuit

In a companion paper [5] we resolved the question of whether cyclic pursuits can exhibit 'non-mutual' captures. Although, as we showed, non-mutual captures can occur, the set of initial conditions which lead to them has Lebesgue measure zero. Thus, generically, cyclic pursuits collapse into a mutual capture. In this paper we consider whether the pursuit configuration can asymptotically approach a regular one for a non-trivial set of initial conditions. More precisely, we study the stability of regular geometries of cyclic pursuit. We show that in all dimensions the only stable regular n-bug shapes are the regular two dimensional n-gons, n greater than or equal to 7, in which each vertex chases its neighboring vertex in some fixed orientation. We also analize the three bug cyclic pursuit in detail, proving that, except for the equilateral initial position, the triangle formed is asymptotically degenerate with the minimum interior angle tending to zero while the vertex at which the minimum is located rotates among the vertices infinitely often.