A Theorem of Global Asymptotical Stability

Given a vector eld X on the real plane, we study the innuence of the curvature of the orbits of _ x = X ? (x) in the stability of those of the system _ x = X(x). We pay special attention to the case in which this curvature is negative in the whole plane. Under this assumption, we classify the possible critical points and give a criterion for a point to be globally asymptotically stable. In the general case, we also provide expressions for the rst three derivatives of the Poincar e map associated to a periodic orbit in terms of geometrical quantities. These results are extracted from the paper GGG].