Calculating periodic orbits of the Hénon–Heiles system - Alhowaity, Sawsan; Abouelmagd, Elbaz I.; Diab, Zouhair; Guirao, Juan L. G. (Frontiers in Astronomy and Space Sciences, 2023)
This work is divided to two parts, the first part is devoted to analyze the features of Hénon--Heiles's potential, and finding the levels energy for bounded and unbounded motions. The critical points are explored in different phase spaces from the classical potential to the generalized one. While in the second part, the possible solutions of generalized (fifth--degree) Hénon--Heiles system are analyzed by using the Averaging Theory. Two consequent transformations are used to set the Hamiltonian of this system in standard form for applying the Averaging Theory. In this context, eight solutions are found, but one of them is not convenient with the proposed assumptions, while the other seven solutions are proper and adequate to represent seven periodic orbits for the generalized Hénon--Heiles dynamical system. Namely, this system has at least seven periodic orbits.