September  2006, 5(3): 537-550. doi: 10.3934/cpaa.2006.5.537

On a criterium of global attraction for discrete dynamical systems

1. 

Dept. de Matemàtiques i Informàtica, Universitat de les Illes Balears, Escola Politècnica Superior, 07122-Palma de Mallorca, Spain, Spain

2. 

Dept. de Matemàtiques, Universitat Autónoma de Barcelona, Edifici C, 08193 Bel-laterra, Barcelona

Received  April 2005 Revised  January 2006 Published  June 2006

Consider that the origin is a fix point of a discrete dynamical system $x^{(n+1)}=F(x^{(n)})$, defined in the whole $\mathbb R^m.$ LaSalle, in his book of 1976, [13], proposes to study several conditions which might imply global attraction. One of his suggestions is to write $F(x)=A(x)x$, where $A(x)$ is a real $m\times m$ matrix, and to assume that all the eigenvalues of eigenvalues of $A(x)$, for all $x\in \mathbb R^m$, have modulus smaller than one. In the paper [4], Cima et al. show that, when $m\ge2$, such hypothesis does not guarantee that the origin is a global attractor, even for polynomial maps $F$. From the observation that the decomposition of $F(x)$ as $A(x)x$ is not unique, in this paper we wonder whether LaSalle condition, for a special and canonical choice of $A,$ forces the origin to be a global attractor. This canonical choice is given by $A_c(x)=\int_0^1 DF(sx) ds,$ where the integration of the matrix $DF(x)$ is made term by term. In fact, we prove that LaSalle condition for $A_c(x)$ is a sufficient condition to get the global attraction of the origin when $m=1,$ or when $m=2$ and $F$ is polynomial. We also show that this is no more true for $m=2$ when $F$ is a rational map or when $m\ge3.$ Finally we consider the equivalent question for ordinary differential equations.
Citation: B. Coll, A. Gasull, R. Prohens. On a criterium of global attraction for discrete dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (3) : 537-550. doi: 10.3934/cpaa.2006.5.537
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