Bifurcations of Periodic Orbits in Axisymmetric Scalefree Potentials

Details Bifurcations of Periodic Orbits in Axisymmetric Scalefree Potentials Bifurcations of Periodic Orbits in Axisymmetric Scalefree Potentials

Sep 1998

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998dda....30.0102h&link_type=abstract

American Astronomical Society, DDA meeting #30, #01.02; Bulletin of the American Astronomical Society, Vol. 30, p.1139

Astronomy and Astrophysics

Astronomy

Scientific paper

We have surveyed orbits in potentials with central cusps, concentrating on QZMH potentials (Qian et al 1995, MN 274, 602) generated by mass distributions with spheroidal equidensity surfaces, and Evans potentials (1994, MN 267, 333) whose equipotentials are spheroidal. Both potentials form two-parameter families; the parameters being the power -beta of dependence on distance, and the axis ratio q of the spheroidal surfaces. Many surfaces of section are regular, showing only tube orbits and no prominent bifurcations. The most prominent bifurcations are due to 1-1 and 4-3 resonances between radial motions and motions perpendicular to the equatorial plane. The 1-1 resonance, which causes the equatorial plane orbit to become unstable, is less common in QZMH than in Evans potentials. A 1-1 resonance instability of the thin tube orbit occurs only in potentials so strongly flattened as not to be relevant for elliptical galaxies. The most intriguing and least understood bifurcations are the period tripling ones. They are also the most common for QZMH potentials. Stable period-3 orbits do not generally arise through a bifurcation from the thin tube at a 120(deg) rotation angle. They can arise de novo in stable and unstable pairs via a turning-point bifurcation, and also through a bifurcation of the equatorial plane orbit. In either case they may be unconnected with the thin tube. To elucidate the genesis of some period-3 orbits, we studied a class of analytic mappings in the neighborhood of an elliptic fixed point. They are derived from a generating function, which guarantees area-preservation and time reversibility, These mappings also form a two-parameter family, one being the rotation angle. We find that there may never be any stable period-3 orbits, and, when there are, they typically have only a short span of existence before becoming unstable through a supercritical pitchfork bifurcation corresponding to a period-doubling instability.

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