title: Bifurcations of periodic orbits in axisymmetric scalefree potentials.
abstract: 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.
authors.affiliation: Department of Mathematics, Florida State University, Tallahassee 32306-4510, USA.
authors.forename: C
authors.initials: C
authors.lastname: Hunter
authors.forename: B
authors.initials: B
authors.lastname: Terzić
authors.forename: A M
authors.initials: AM
authors.lastname: Burns
authors.forename: D
authors.initials: D
authors.lastname: Porchia
authors.forename: C
authors.initials: C
authors.lastname: Zink