A note on LaSalle's problems

Anna Cima¹ ; Armengol Gasull¹ ; Francesc Mañosas¹

¹ Dept. de Matemàtiques Universitat Autònoma de Barcelona Edifici Cc, 08193 Bellaterra, Barcelona, Spain

Annales Polonici Mathematici, Tome 76 (2001) no. 1-2, pp. 33-46

Résumé

In LaSalle's book “The Stability of Dynamical Systems", the author gives four conditions which imply that the origin of a discrete dynamical system defined on $ {\mathbb R}$ is a global attractor, and proposes to study the natural extensions of these conditions in $ {\mathbb R}^n.$ Although some partial results are obtained in previous papers, as far as we know, the problem is not completely settled. In this work we first study the four conditions and prove that just one of them implies that the origin is a global attractor in $ {\mathbb R}^n$ for polynomial maps. Then we note that two of these conditions have a natural extension to ordinary differential equations. One of them gives rise to the well known Markus–Yamabe assumptions. We study the other condition and we prove that it does not imply that the origin is a global attractor.

Détail

DOI : 10.4064/ap76-1-4

Keywords: lasalles book stability dynamical systems author gives conditions which imply origin discrete dynamical system defined mathbb global attractor proposes study natural extensions these conditions mathbb although partial results obtained previous papers far know problem completely settled work first study conditions prove just implies origin global attractor mathbb polynomial maps note these conditions have natural extension ordinary differential equations gives rise known markus yamabe assumptions study other condition prove does imply origin global attractor