Systems of transversal sections near critical energy levels of hamiltonian systems in $Mathbb{R}^{4}$

In this article the authors study Hamiltonian flows associated to smooth functions H:\mathbb R^4 \to \mathbb R restricted to energy levels close to critical levels. They assume the existence of a saddle-center equilibrium point p_c in the zero energy level H^{-1}(0). The Hamiltonian function near p_c is assumed to satisfy Moser's normal form and p_c is assumed to lie in a strictly convex singular subset S_0 of H^{-1}(0). Then for all E \gt 0 small, the energy level H^{-1}(E) contains a subset S_E near S_0, diffeomorphic to the closed 3-ball, which admits a system of transversal sections \mathcal F_E, called a 2-3 foliation. \mathcal F_E is a singular foliation of S_E and contains two periodic orbits P_2,E\subset \partial S_E and P_3,E\subset S_E\setminus \partial S_E as binding orbits. P_2,E is the Lyapunoff orbit lying in the center manifold of p_c, has Conley-Zehnder index 2 and spans two rigid planes in \partial S_E. P_3,E has Conley-Zehnder index 3 and spans a one parameter family of planes in S_E \setminus \partial S_E. A rigid cylinder connecting P_3,E to P_2,E completes \mathcal F_E. All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to P_2,E in S_E\setminus \partial S_E follows from this foliation.