AUTHOR=Alhowaity Sawsan , Abouelmagd Elbaz I. , Diab Zouhair , Guirao Juan L. G. 

TITLE=Calculating periodic orbits of the Hénon–Heiles system

JOURNAL=Frontiers in Astronomy and Space Sciences

VOLUME=Volume 9 - 2022

YEAR=2023

URL=https://www.frontiersin.org/journals/astronomy-and-space-sciences/articles/10.3389/fspas.2022.945236

DOI=10.3389/fspas.2022.945236

ISSN=2296-987X

ABSTRACT=This work is divided to two parts, the first part is devoted to analyze the features of H\'{e}non--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)  H\'{e}non--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  H\'{e}non--Heiles dynamical system. Namely, this system has at least seven periodic orbits.