An Exploration of the Topological Structure and Bifurcation of Liouville Tori in Models of Galactic Motion

This paper examines two integrable cases: a generalized Hénon-Heiles (HH) system and a quartic potential. For each case, the Liouville tori's bifurcation (LTB) is analyzed. Periodic solutions (PS) are derived using Jacobi elliptic functions, and the corresponding phase portraits are presented with a classification of the singular points. Furthermore, the PS for both cases are constructed based on the Lyapunov theorem. The possible applications of this study are primarily confined to celestial mechanics and astrodynamics. Specifically, the generalization of the HH system and quartic potentials often appears in modeling gravitational interactions between celestial bodies, including the study of the stability and motion of planets, asteroids, and satellites. Additionally, the classification of singular points and phase portraits provides valuable insights for identifying chaotic or regular behaviors in various systems, such as weather models or economic systems. Furthermore, understanding bifurcations and PS contributes to the design of control mechanisms for nonlinear systems, including robotics and automated processes.