A novel approach for the stability problem in non-linear dynamical systems. We present a methodology for the determination of sufficient conditions for the existence of a Lyapunov function in a general class of non-linear dynamical systems. The algorithm can be applied in the cases where the system parameters are numerically specified or not. The numerical algorithm involves the resolution of a linear programming problem. The algebraic version is implemented using the MAPLE programming system in the package Lyapunov.