标题： 环动力学中共振的稳定性和分岔 显示英文标题

标题： Stability and bifurcations of resonances in ring's dynamics

摘要： 我们使用摄动理论和分岔理论来分析与描述一个粒子在天体周围的环中运动的模型相关的共振的动力学行为。 中心天体被建模为一个均匀的三轴椭球体，绕其最短物理轴以恒定的角速度旋转。 假设无质量的环状粒子位于椭球体的赤道平面内。 通过应用附振变量来研究粒子的动力学，这使得共转共振和林德布拉德共振的定义变得直接明了。 这些共振与一个具有两个自由度的哈密顿函数相关，我们为此计算了正则形式和共振哈密顿量的适当展开式。 最初，验证了正则形式是非退化的，从而保证了不变KAM环面的存在，通过它们在相空间中的限制，提供了共振的稳定性。 随后，检查了两个典型的测试案例：一个接近球形的椭球体和一个高度非球形的椭球体。 此外，本研究集中在三个主要共振上：共转、1:2和1:3，我们展示了通过分析模型的哈密顿表述和共振正则形式得到的这些共振的动力学行为结果。 具体而言，我们研究了相空间结构、共振附近的摆动振幅以及分岔的发生情况。 值得注意的是，在所研究的两个测试案例中，对于偏心率的相关值，1:3共振没有表现出分岔的证据。 因此，我们的动力学研究表明，与其他共振相比，选择$1:3$共振的可能性更大。 显示英文摘要

摘要： We use perturbation theory and bifurcation theory to analyze the dynamical behavior of resonances, associated to a model describing a particle moving within a ring around a celestial object. The central body is modeled as a homogeneous triaxial ellipsoid, rotating about its shortest physical axis at a constant angular velocity. It is assumed that the massless ring particle moves within the equatorial plane of the ellipsoid. The dynamics of the particle is studied using epicyclic variables, that lead to a straightforward definition of corotation and Lindblad resonances. These resonances are associated to a Hamiltonian function with two degrees of freedom, for which we compute appropriate expansions for the normal form and the resonant Hamiltonian. Initially, the normal form is verified to be non--degenerate, thereby guaranteeing the existence of invariant KAM tori, providing the stability of the resonances, through their confinement in phase space. Subsequently, two exemplary test cases are examined: a nearly spherical ellipsoid and a highly aspherical ellipsoid. Furthermore, this study concentrates on three principal resonances: corotation, 1:2, and 1:3, for which we present results concerning their dynamical behavior obtained analyzing the Hamiltonian formulation of the model and the resonant normal form. Specifically, we examine the phase space structure, the amplitude of libration around the resonances, and the occurrence of bifurcations. Remarkably, in none of the two studied test cases the 1:3 resonance presents evidence of bifurcations for relevant values of the eccentricity. Our dynamical study thus supports a greater probability of selecting the $1:3$ resonance in comparison to the other resonances.