轴对称无标度势中的共振与分岔
Resonances and bifurcations in axisymmetric scale-free potentials
摘要：
我们调查的分叉的共振“薄”管在轴对称星系势的家庭的分析处理。我们验证，最相关的分叉是由于（1：1）共振产生的“倾斜”的轨道，通过两种不同的机制：从磁盘轨道和从“薄”管相关的垂直振荡。在这些之后发生的最接近的共振是扁长情况下的（4：3）共振和扁长情况下的（2：1）共振。（1：1）共振处理在一个简单的方式使用2阶截断范式。高阶共振的研究反而很麻烦，因为标准形必须被截断到很高的程度，而且项的数量增长非常快。因此，我们采用了进一步的简化给出的参数值的解析公式，在分叉随之而来，并比较它们与选定的数值结果。由于所涉及的级数的渐近性质，预测的可靠性远远超过了原始级数的收敛半径。
英文摘要：
We investigate an analytical treatment of bifurcations of families of resonant `thin' tubes in axisymmetric galactic potentials. We verify that the most relevant bifurcations are due to the (1:1) resonance producing the `inclined' orbits through two different mechanisms: from the disk orbit and from the `thin' tube associated to the vertical oscillation. The closest resonances occurring after these are the (4:3) resonance in the oblate case and the (2:1) resonance in the prolate case. The (1:1) resonances are treated in a straightforward way using a 2nd-order truncated normal form. The higher-order resonances are instead cumbersome to investigate, because the normal form has to be truncated to a high degree and the number of terms grows very rapidly. We therefore adopt a further simplification giving analytic formulas for the values of the parameters at which bifurcations ensue and compare them with selected numerical results. Thanks to the asymptotic nature of the series involved, the predictions are reliable well beyond the convergence radius of the original series.
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