关于三体问题中周期轨道的分支与延拓
On the bifurcation and continuation of periodic orbits in the three-body problem
摘要：
本文研究了行星型平面三体问题周期轨道的生成与延拓，主要研究了非对称周期轨道的生成与延拓。非对称轨道只存在于被称为“非对称共振”的特殊共振中。然而，数值研究表明，在一般的三体问题中，不对称轨道可能不仅存在于非对称共振，但对其他类型的，太。本文证明了椭圆型限制问题中非对称周期轨道的存在性。这些轨道是连续的，澄清了一般问题中许多非对称周期轨道的起源。此外，我们还说明了限制圆问题的周期轨道族和椭圆问题的周期轨道族是如何平滑地连接在一起并在一般问题中形成族的，以这种方式验证了Bozis和Hadjidemetriou（1976）首先描述的情况。
英文摘要：
We consider the planar three body problem of planetary type and we study the generation and continuation of periodic orbits and mainly of asymmetric periodic orbits. Asymmetric orbits exist in the restricted circular three body problem only in particular resonances called "asymmetric resonances". However, numerical studies showed that in the general three body problem asymmetric orbits may exist not only for asymmetric resonances, but for other kinds, too. In this work, we show the existence of asymmetric periodic orbits in the elliptic restricted problem. These orbits are continued and clarify the origin of many asymmetric periodic orbits in the general problem. Also, we illustrate how the families of periodic orbits of the restricted circular problem and those of the elliptic one join smoothly and form families in the general problem, verifying in this way the scenario described firstly by Bozis and Hadjidemetriou (1976).
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