Bifurcations of periodic orbits in axisymmetric scalefree potentials.

We study orbits in potentials with central cusps, emphasizing the spheroidal equidensity (SED) potentials generated by mass distributions with spheroidal equidensity surfaces. The most prominent bifurcations are those related to 1:1 and 4:3 resonances between radial motions and motions perpendicular to the central plane. We find that 1:1 resonances can cause the thin tube orbit, as well as the equatorial plane orbit, to become unstable. We concentrate on period-tripling bifurcations because they appear to be the least understood. We study them via a class of analytic maps. This study suggests that stable period-three orbits generally arise de novo in stable and unstable pairs via a turning-point bifurcation, and not through a bifurcation from the thin tube at a 120 degree rotation angle. The stable period-three orbits typically have only a short span of existence before becoming unstable to a period-doubling instability through a supercritical pitchfork bifurcation.