Some Results About Global Asymptotic Stability

• Álvaro Castañeda ^[1] ; Víctor Guíñez ^[2]
  1. [1] Universidad de Chile

     Universidad de Chile

     Santiago, Chile

     
  2. [2] Universidad de Santiago de Chile

     Universidad de Santiago de Chile

     Santiago, Chile

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 12, Nº 2, 2013, págs. 427-441
• Idioma: inglés
• DOI: 10.1007/s12346-013-0102-8
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• Resumen

  • 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.