Some Results About Global Asymptotic Stability

Abstract

We study the global asymptotic stability of the origin for the continuous and discrete dynamical system associated to polynomial maps in ℝn (especially when n = 3) of the form F = λ I + H, with F(0) = 0, where λ 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 ℝ and when they are linearly independent over ℝ. 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).