Some Results About Global Asymptotic Stability

We study the global asymptotic stability of the origin for the continuous and discrete dynamical system associated to polynomial maps in ?n (especially when n = 3) of the form F = ? I + H, with F(0) = 0, where ? is a real number, I the identity map, and H a map with nilpotent Jacobian matrix J H. We distinguish the cases when the rows of J H are linearly dependent over ? and when they are linearly independent over ?. In the linearly dependent case we find non-linearly triangularizable vector fields F for which the origin is globally asymptotically stable singularity (respectively fixed point) for continuous (respectively discrete) systems generated by F. In the independent continuous case, we present a family of maps that have orbits escaping to infinity. Finally, in the independent discrete case, we show a large family of vector fields that have a periodic point of period 3. © 2013 The Author(s).