Toroidal Periodic-Orbit Dividing Surfaces in Hamiltonian Systems with Four Degrees of Freedom: Geometric Construction and the No-Recrossing Property

In this paper, we extend a periodic-orbit-based method for the construction of dividing surfaces to Hamiltonian systems with four degrees of freedom. Dividing surfaces are fundamental phase space structures that separate regions associated with different dynamical behavior and are essential for the study of transport and reaction dynamics. In higher-dimensional systems, their computation is typically based on Normally Hyperbolic Invariant Manifolds (NHIMs), whose determination becomes increasingly demanding as the dimensionality of the system increases. The method employed in this work constructs dividing surfaces using periodic orbits, avoiding the direct computation of NHIMs. We consider a quadratic normal form Hamiltonian system with four degrees of freedom possessing an index-1 saddle and develop a systematic procedure for constructing dividing surfaces in the corresponding seven-dimensional energy surface. The resulting objects exhibit a higher-dimensional toroidal geometry, arising from the combination of periodic orbit projections with auxiliary geometric structures. We analyze the properties of these dividing surfaces and establish conditions under which the local no-recrossing property with respect to the reaction coordinate is satisfied. Our results demonstrate that the periodic-orbit-based approach provides an effective and scalable framework for studying phase space transport in higher-dimensional Hamiltonian systems.