Naiara Vergian de Paulo Costa

Abstract. In this talk, I will discuss the dynamics of a two-degree-of-freedom
mechanical system by using tools from Symplectic Dynamics. To be more pre-
cise, we consider a potential function on the plane with finitely many saddle
points at the same critical level. As the energy increases across the critical
value, a disk-like component of the Hill region gets connected to other com-
ponents precisely at the saddles. Under certain convexity assumptions on the
critical set, we obtain a genus zero transverse foliation in a region of the energy
surface where an interesting dynamics takes place. This singular foliation is
obtained by means of the theory of pseudo-holomorphic curves. Its binding set
is formed by finitely many periodic orbits, including the Lyapunov orbits in
the neck region about the rest points, and its regular leaves consist of planes
and cylinders transverse to the flow. The transverse foliation forces the exis-
tence of infinitely many periodic orbits, homoclinics, and heteroclinics to the
Lyapunov orbits in case their actions coincide. We apply the results to the
Hénon-Heiles system and to the Euler problem of two fixed centers. This is a
joint work with Seongchan Kim, Pedro Salomão, and Alexsandro Schneider.