Irregular Period Tripling Bifurcations in Axisymmetric Scalefree Potentials

Details Irregular Period Tripling Bifurcations in Axisymmetric Scalefree Potentials Irregular Period Tripling Bifurcations in Axisymmetric Scalefree Potentials

Sep 1998

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998dda....30.0103t&link_type=abstract

American Astronomical Society, DDA meeting #30, #01.03; Bulletin of the American Astronomical Society, Vol. 30, p.1139

Astronomy and Astrophysics

Astronomy

Scientific paper

Irregular Period Tripling Bifurcations in Axisymmetric Scalefree Potentials We have examined a range of orbits in axisymmetric potentials with central cusps, focusing on QZMH potentials (Qian et al 1995, MN 274, 602) which arise from mass distribution with spheroidal equidensity surfaces, Evans potentials (1994, MN 267, 333) with spheroidal equidensity surfaces and logarithmic potentials as a special case of Evans potentials. Both QZMH and Evans potentials are characterized by two parameters - the power -beta of dependence on distance, and the flatness parameter (axis ratio) q. Logarithmic potential is a special case of the Evans potential when beta =0. 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 all three of the potentials. The period-3 orbits which bifurcate in stable/unstable pairs via pitchfork bifurcation from the stable thin tube orbit at a 120(deg) 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, de novo, 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 / 3} line. Invariant curves surrounding the stable period-3 cycles are manifested through the flattenings of the curve at rot={1 / 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).

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