Calculating periodic orbits of the Henon-Heiles system

This work is divided to two parts; the first part analyzes the features of Henon-Heiles's potential and finding the energy levels for bounded and unbounded motions. The critical points are explored in different phase spaces from the classical potential to the generalized one. In the second part, the possible solutions of the generalized (fifth-degree) Henon-Heiles system are analyzed 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, where one of them is not convenient for the proposed assumptions, and the other seven solutions are proper and adequate to represent seven periodic orbits for the generalized Henon-Heiles dynamical system, which has at least seven periodic orbits.