Narrow Results

In two previous papers [Katsanikas & Wiggins, 2021a, 2021b], we developed two methods for the construction of periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom. We applied the first method (see [Katsanikas & Wiggins, 2021a]) in the case of a quadratic Hamiltonian system in normal form with three degrees of freedom, constructing a geometrical object that is the section of a 4D toroidal structure in the 5D energy surface with the space $x=0$. We provide a more detailed geometrical description of this object within the family of 4D toratopes. We proved that this object is a dividing surface and it has the no-recrossing property. In this paper, we extend the results for the case of the full 4D toroidal object in the 5D energy surface. Then we compute this toroidal object in the 5D energy surface of a coupled quadratic normal form Hamiltonian system with three degrees of freedom.

Motivated by Poincaré’s orbits going to infinity in the (restricted) three-body problem [see H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 3 (Gauthier-Villars, 1899) and A. Chenciner, Poincaré and the three-body problem, in Henri Poincaré, 1912–2012 (Birkhäuser, Basel, 2015), pp. 51–149], we investigate the generic existence of heteroclinic-like orbits in a neighborhood of the critical set of a $b$-contact form. This is done by using a singular counterpart [R. Cardona, E. Miranda and D. Peralta-Salas, Euler flows and singular geometric structures, Philos. Trans. R. Soc. A 377(2158) (2019) 20190034] of Etnyre–Ghrist’s contact/Beltrami correspondence [J. Etnyre and R. Ghrist, Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture, Nonlinearity 13(2) (2000) 441–458], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [Generic properties of eigenfunctions, Amer. J. Math. 98(4) (1976) 1059–1078]. Specifically, we analyze the $b$-Beltrami vector fields on $b$-manifolds of dimension $3$ and prove that for a generic asymptotically exact $b$-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric $b$-Beltrami vector field on an asymptotically flat $b$-manifold has a generalized singular periodic orbit and at least four escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose $\alpha$- and $\omega$-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture.