Note on the Markus-Yamabe conjecture for gradient dynamical systems

Let upsilon : R-n -> R-n be a C-1 vector field which has a singular point O and its linearization is asymptotically stable at every point of R-n. We say that the vector field v satisfies the Markus-Yamabe conjecture if the critical point O is a global attractor of the dynamical system x = upsilon(x). In this note we prove that if upsilon is a gradient vector field, i.e. upsilon = del f (f is an element of C-2), then the basin of attraction of the critical point O is the whole R-n, thus implying the Markus-Yamabe conjecture for this class of vector fields. An analogous result for discrete dynamical systems of the form x(m+1) = del f (x(m)) is proved. (c) 2005 Elsevier Inc. All rights reserved.