Hunter C, Terzić B, Burns A M, Porchia D, Zink C
Department of Mathematics, Florida State University, Tallahassee 32306-4510, USA.
Ann N Y Acad Sci. 1998 Dec 30;867:61-84. doi: 10.1111/j.1749-6632.1998.tb11250.x.
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.
我们研究具有中心尖点势场中的轨道，重点关注由具有椭球等密度面的质量分布所产生的椭球等密度（SED）势场。最显著的分岔是那些与径向运动和垂直于中心平面的运动之间的1:1和4:3共振相关的分岔。我们发现1:1共振会导致细管轨道以及赤道平面轨道变得不稳定。我们专注于周期三倍分岔，因为它们似乎是理解最少的。我们通过一类解析映射来研究它们。这项研究表明，稳定的周期三轨道通常通过转折点分岔在稳定和不稳定对中从头出现，而不是通过细管在120度旋转角度处的分岔。稳定的周期三轨道通常在通过超临界叉形分岔对周期加倍不稳定性变得不稳定之前，只有很短的存在跨度。
