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 ℝm. 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 × m matrix, and to assume that all the eigenvalues of eigenvalues of A(x), for all x ε ℝm, have modulus smaller than one. In the paper [4], Cima et al. show that, when m ≥ 2, 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 Ac(x) = f10 DF(sx) ds, where the integration of the matrix DF(x) is made term by term. In fact, we prove that LaSalle condition for Ac (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 ≥ 3. Finally we consider the equivalent question for ordinary differential equations.