论文标题
在失调的2：4共振上
On the detuned 2:4 resonance
论文作者
论文摘要
我们认为，在1：2共振的情况下，以两种自由度的汉密尔顿系统家族均具有平衡。在引人注目的情况下，这种“费米共振”通常会导致正常模式通过倍增分叉而失去其稳定性。对于立方电位，这涉及短轴向轨道和银河系动力学，所得稳定的周期轨道称为“香蕉”轨道。银河电位相对于坐标平面是对称的，因此电势和正常形式都没有立方术语。这个$ \ mathbb {z} _2 \ times \ times \ mathbb {z} _2 $ -smmetry将1：2共振变成高阶共振，因此也说2：4共振。在本文中，我们本身研究了2：4共振，而不仅限于自然的哈密顿系统，其中$ h = t + v $由动力学和（位置）势能组成。然后，短轴向轨道在各处都是动态稳定的，除了在香蕉和“抗冰淇淋”轨道的同时分叉，而现在是长轴向轨道，通过两次连续的周期性加倍分叉，可以失去并恢复稳定性。
We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials this concerns the short axial orbits and in galactic dynamics the resulting stable periodic orbits are called "banana" orbits. Galactic potentials are symmetric with respect to the co-ordinate planes whence the potential -- and the normal form -- both have no cubic terms. This $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetry turns the 1:2 resonance into a higher order resonance and one therefore also speaks of the 2:4 resonance. In this paper we study the 2:4 resonance in its own right, not restricted to natural Hamiltonian systems where $H = T + V$ would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and "anti-banana" orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.
