Systems of Transversal Sections near Critical Energy Levels of Hamiltonian Systems in R⁴ - Paulo, Naiara V. de, 1986- (Memoirs of the American Mathematical Society Ser., 2018)
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 \mathc.