Systems of transversal sections near critical energy levels of Hamiltonian systems in ${\mathbb{R}}^4$

In this article we study Hamiltonian flows associated to smooth functions $H:{\mathbb{R}}^4\to \mathbb{R}$ restricted to energy levels close to critical levels. We 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 $E>0$ 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$P_{3,E}\subset S_E\setminus \partial S_E$asbindingorbits\mathrm{.}$P_{2,E}$istheLyapunofforbitlyinginthecentermanifoldof$p_c$, has Conley-Zehnder index $2$ and spans two rigid planes in $\mathrm{\partial }{S}_E$. $P_{3,E}$ has Conley-Zehnder index $3$ and spans a one parameter family of planes in $S_E\setminus \mathrm{\partial }{S}_E$. 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 $S_E\setminus \mathrm{\partial }{S}_E$ follows from this foliation.