3D Generating Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – I

In our earlier research [ Katsanikas & Wiggins , 2021a , 2021b , 2023a , 2023b , 2024a , 2024b , 2024c ], we developed two new approaches for building up dividing surfaces in the phase space of Hamiltonian systems with three or more degrees of freedom. These surfaces were derived either from periodic orbits or from 2D or 3D generating surfaces in the phase space. Our previous work extended the realization of these dividing surfaces into more intricate forms, such as tori or cylinders, situated within the constant energy manifold of the Hamiltonian system. In those studies, we utilized the above-mentioned surfaces in the setting of a three-degrees-of-freedom quadratic normal form Hamiltonian system. This series of papers extends our findings to 3D generating surfaces for a three-degrees-of-freedom quartic normal form Hamiltonian system. The focus of this paper is to introduce the first method for constructing 3D generating surfaces.