Abstract A signi cant number of problems arising in science and engineering can be modeled by a system of PDE's, which after discretization in space leads to non linear dynamical systems of the form G(x; alpha) = 0 depending on a parameter alpha. In this work we analize some Krylov methods as GMRES (Generalized Minimal Residual) and IRA (Implicitly Restarted Arnoldi) when they are imbbeded in a continuation framework to compute solutions of the parameterized non linear system G(x; alpha) = 0, for different values of the parameter alpha. By using these techniques it is possible to detect and compute important steady states which give us information about the stability of the modeled process. Experiments on several test problems reveal the reliability of this approach in the accurate detection of critical and bifurcation points. Additionally, stability in continuation points is obtained at no cost.