Estudo da dinâmica local das aplicações de Hénon cúbicas conservativas

Abstract

The goal of the present work is to show a study of local aspects of the dynamics from the family of conservative cubic Hénon maps in a neighbourhood of its fixed points, that could be hyperbolic, reversed hyperbolic, elliptic or parabolic. The definitions of stability and instability used are given by J. K. Moser. On the hyperbolic and reversed hyperbolic cases, it will be proven that every diffeomorphism is unstable in its fixed point, using the Hartman–Grobman Theorem. It will be shown, on the first case, the existence of topologically transverse homoclinic points for a large parcel of the family in question. Therefore, the maps that satisfies this condition are chaotics near the fixed point. On the elliptic case, it will be proven that the maps are stable in all its fixed points non-resonants until sixth order, where the Birkhoff’s Normal Form and the Moser’s Twist Theorems are used. For the parabolic case, it will be shown that the maps of the referred family are unstable on the fixed point according to a criterion established by T. Levi-Civita, with the exception of two of them, in which case there is no statement about its stability.