This paper considers the identification problem of nonlinear systems based on single-hidden-layer neural networks (SHLNNs) and Lyapunov theory. A nonlinearly parameterized neural model, whose weights are adjusted by robust adaptive laws, which are designed via Lyapunov theory, is proposed for ensuring the convergence of the residual state error to an arbitrary neighborhood of zero. In addition, a scaling matrix is used to resize the unknown nonlinearities to be approximated by an SHLNN, which, in turn, provides a simple way to shape the residual state error. It is shown that all estimation errors are uniformly bounded and, in addition, that the residual state error is uniformly ultimately bounded with an ultimate bound that depends directly on some independent design parameters. To validate the theoretical results, the identification of a chaotic system and a comparison study with other work in the literature are performed.