On the multiplicity of periodic orbits and homoclinics near critical energy levels of Hamiltonian systems in R⁴

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We study two-degree-of-freedom Hamiltonian systems. Let us assume that the zero energy level of a real-analytic Hamiltonian function H:R⁴ → R contains a saddle-center equilibrium point lying in a strictly convex sphere-like singular subset S₀⊂ H⁻¹(0). From previous work [de Paulo-Salomão, Memoirs of the AMS] we know that for any small energy E>0, the energy level H⁻¹(E) contains a closed 3-ball Sₑ in a neighborhood of S₀ admitting a singular foliation called 2-3 foliation. One of the binding orbits of this singular foliation is the Lyapunoff orbit P₂,ₑ contained in the center manifold of the saddle-center. The other binding orbit lies in the interior of Sₑ and spans a one parameter family of disks transverse to the Hamiltonian vector field. In this article we show that the 2-3 foliation forces the existence of infinitely many periodic orbits and infinitely many homoclinics to P₂,ₑ in Sₑ. Moreover, if the branches of the stable and unstable manifolds of P₂,ₑ inside Sₑ do not coincide then the Hamiltonian flow on Sₑ has positive topological entropy. We also present applications of these results to some classical Hamiltonian systems.