Irregular Period Tripling bifurcations in axisymmetric scalefree potentials

B. Terzic

We have examined a range of orbits in axisymmetric potentials with central cusps, focusing on SED potentials (Qian et al 1995, MN 274, 602) which arise from mass distribution with spheroidal equidensity surfaces (hence the acronim) The SED potentials are characterized by two parameters - the power $-\beta$ of dependence on distance, and the flatness parameter (axis ratio) $q$. Our primary focus is on understanding period tripling bifurcations in these potentials, which is related to 4:3 resonances between radial and vertical frequencies.

The period tripling bifurcations are observed in these potentials. The period-3 orbits which bifurcate in stable/unstable pairs via pitchfork bifurcation from the stable thin tube orbit at a $120^{\circ}$ rotation angle at some critical $L_z^2$ are called regular. However, of particular interest to us are "irregular" orbits which bifurcate away from the thin tube, via a turning-point bifurcation.

In order to understand this phenomenon, we have analyzed the rotation numbers ($rot$) of the invariant curves in the $(R,\dot R)$ phase space. Our numerical results reveal the nature of the rotation curve which leads to these irregular bifurcations. Period-3 orbits bifurcate when this rotation curve crosses $rot = {1 \over 3}$ line. Invariant curves surrounding the stable period-3 cycles are manifested through the flattenings of the curve at $rot={1 \over 3}$. Also, higher order resonances are observed in the outer regions, as manifested through smaller flattenings of the rotation curve (as the invariant curves surrounding them are less dominant).