Sobre a existência de pontos homoclínicos em vizinhanças de pontos fixos elípticos

Abstract

In this dissertation, we study certain aspects of symplectic diffeomorphisms in the plane near an elliptic fixed points.Our approach considers a Birkhoff formal norm to ensure the existence ofhyperbolic and elliptic periodic orbits in every neighborhood of the ellipticfixed point and it will be shown that the set of real analytic symplectic diffeomorphisms with elliptic fixed point at zero and have elliptical orbits and hyperbolic in every neighborhood of zero is a residual subset.We will develop the theory of local invariant manifolds, we get a representation of the invariant manifolds as graph functions. Thus, you can ensure the proximity of the stable and unstable varieties two hyperbolic points in the same orbit. Finally, using the fact that diffeomorphism preserves area, prove that the stable and unstable varieties actually have the intersection point, thus obtainingan called homoclinic points. The main reference is a paper of Edward Zehnder [1] \Homoclinic points near elliptic fixed points".