A simple test for asymptotic stability in some dynamical systems.

Abstract

In this paper we analyze asymptotic stability of the dynamical system x˙ = f(x) defined by a C1 function f : D Rn where D Rn + is an open set. We obtain a criterion of stability for the equilibrium solution ¯x D when the vector field f satisfies a) ifi(¯x) < 0 and b) ( ifi(¯x))−1 ifj(¯x) + ( jfj(¯x))−1 jfi(¯x) > 2 for i, j = 1, . . . ,n.