2D Generating Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – I

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In previous studies, we developed two techniques aimed at expanding the scope of constructing a periodic orbit dividing surface within a Hamiltonian system with three or more degrees of freedom. Our approach involved extending a periodic orbit into a torus or cylinder, thereby elevating it into a higher-dimensional entity within the energy surface (see [ Katsanikas & Wiggins , 2021a , 2021b , 2023a , 2023b ]). Recently, we introduced two alternative methods for creating dividing surfaces, distinct from the utilization of periodic orbits, by employing 2D surfaces (geometric entities) or 3D surfaces within a Hamiltonian system with three degrees of freedom (refer to [ Katsanikas & Wiggins , 2024a , 2024b , 2024c ]). In these studies, we applied these surfaces in a quadratic normal form Hamiltonian system with three degrees of freedom. In this series of two papers, we extend our results to 2D generating surfaces for quartic Hamiltonian systems with three degrees of freedom. This paper presents the first method of constructing 2D generating surfaces.