Note on the Markus–Yamabe conjecture for gradient dynamical systems

Abstract

Let v : 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 ˙ = v ( x ) . In this note we prove that if v is a gradient vector field, i.e. v = ∇ f ( f ∈ 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 = ∇ f ( x m ) is proved.