Title: Bifurcations of Riemann Ellipsoids Show Chinese title

Title: 黎曼椭球的分支

Abstract: We give an account of the various changes in the stability character in the five types of Riemann ellipsoids by establishing the occurrence of different quasi-periodic Hamiltonian bifurcations. Suitable symplectic changes of coordinates, that is, linear and non-linear normal form transformations are performed, leading to the characterisation of the bifurcations responsible of the stability changes. Specifically we find three types of bifurcations, namely, Hamiltonian pitchfork, saddle-centre and Hamiltonian-Hopf in the four-degree-of-freedom Hamiltonian system resulting after reducing out the symmetries of the problem. The approach is mainly analytical up to a point where non-degeneracy conditions have to be checked numerically. We also deal with the regimes in the parametric plane where Liapunov stability of the ellipsoids is accomplished. This strong stability behaviour occurs only in two of the five types of ellipsoids, at least deductible only from a linear analysis. Show Chinese abstract

Abstract: 我们通过建立不同拟周期哈密顿分支的发生来描述五种黎曼椭球稳定性特征的各种变化。 进行了适当的辛坐标变换，即线性和非线性正规形式变换，从而刻画了导致稳定性变化的分支。 具体来说，我们在消除了问题的对称性后得到的四自由度哈密顿系统中发现了三种类型的分支，即哈密顿分叉、鞍-中心和哈密顿-霍夫分支。 该方法在需要数值检查非退化条件之前主要是分析性的。 我们还处理了椭球体李雅普诺夫稳定性的参数平面中的区域。 这种强稳定性行为仅发生在五种椭球体中的两种，至少从线性分析中可以得出这一点。