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 over(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.