On a criterium of global attraction for discrete dynamical systems

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.