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The H\'enon-Heiles potential was first proposed as a simplified version ofthe gravitational potential experimented by a star in the presence of agalactic center. Currently, this system is considered a paradigm in dynamicalsystems because despite its simplicity exhibits a very complex dynamicalbehavior. In the present paper, we perform a series expansion up to thefifth-order of a potential with axial and reflection symmetries, which aftersome transformations, leads to a generalized H\'enon-Heiles potential. Such newsystem is analyzed qualitatively in both regimes of bounded and unboundedmotion via the Poincar\'e sections method and plotting the exit basins. On theother hand, the quantitative analysis is performed through the Lyapunovexponents and the basin entropy, respectively. We find that in both regimes thechaoticity of the system decreases as long as the test particle energy gets farfrom the critical energy. Additionally, we may conclude that despite theinclusion of higher order terms in the series expansion, the new system showswider zones of regularity (islands) than the ones present in the H\'enon-Heilessystem.

Dynamical analysis of bounded and unbounded orbits in a generalized H\'enon-Heiles system