Periodic Orbit Dividing Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – II

Abstract

In prior studies [ Katsanikas & Wiggins , 2021a , 2021b , 2023a , 2023b ], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored specifically for Hamiltonian systems with three or more degrees of freedom. These approaches, as described in the aforementioned papers, were applied to a quadratic Hamiltonian system in its normal form with three degrees of freedom. Within this framework, we provide a more intricate geometric characterization of this entity within the family of 4D toratopes which elucidates the structure of the dividing surfaces discussed in these works. Our analysis affirmed the nature of this construction as a dividing surface with the property of no-recrossing. These insights were derived from analytical findings tailored to the Hamiltonian system discussed in these publications. In this series of papers, we extend our previous findings to quartic Hamiltonian systems with three degrees of freedom. We establish the no-recrossing property of the PODS for this class of Hamiltonian systems and explore their structural aspects. Additionally, we undertake the computation and examination of the PODS in a coupled scenario of quartic Hamiltonian systems with three degrees of freedom. In the initial paper [ Gonzalez Montoya et al. ,  2024 ], we employed the first methodology for constructing PODS, while in this paper, we utilize the second methodology for the same purpose.