Note on the Markus–Yamabe conjecture for gradient dynamical systems

Let v :Rn→Rn be a C1 vector field which has a singular point O and its linearization is asymptotically stable at every point of Rn. 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 = v(x). In this note we prove that if v is a gradient vector field, i.e. v = ∇f (f ∈ C2), then the basin of attraction of the critical point O is the whole Rn, thus implying the Markus–Yamabe conjecture for this class of vector fields. An analogous result for discrete dynamical systems of the form xm+1 =∇f (xm) is proved. © 2005 Elsevier Inc. All rights reserved