The objective of this paper is to contribute to the problem of the domain of attraction (DOA) of locally stable non-linear systems. Determining exactly the attraction basin is, in general, a difficult open problem. In our proposal, the main original idea is to combine tools from bifurcation theory and from Lyapunov theory. It is believed that this interplay is the most adequate to get a nice picture of the essential objects affecting local stability. The viability of all these ideas is confirmed by the successfull application to the Furuta pendulum. The strategies proposed here offer the advantage of a conceptually appealing approach, which obtains a conservative, ellipsoidal estimation of the DOA. This ellipsoid is reasonably good and the computational burden is drastically reduced (compared to the pure Lyapunov 'blind search'). The search is improved when bifurcation information is taken into account. The synthesis problem is also considered, and synthesis conditions are given in the form of feasible parameter perturbations which yield an improvement of the DOA estimations.