Homogeneous Lyapunov function for homogeneous continuous vector field

The goal of this article is to provide a construction of a homogeneous Lyapunov function associated with a system of differential equations , under the hypotheses: (1) vanishes at x = 0 and is homogeneous; (2) the zero solution of this system is locally asymptotically stable. Moreover, the Lyapunov function tends to infinity with x, and belongs to , with as large as wanted. As application to the theory of homogeneous systems, we present two well known results of robustness, in a slightly extended form, and with simpler proofs.