A new method is presented for estimating regions of asymptotic stability for autonomous nonlinear systems. The method applies to systems that can be proven to be non-oscillatory, using a suitable Lyapunov technique. It is especially effective for second-order systems that possess several equilibrium states for which the regions of attraction must be found. The determination of the regions of asymptotic stability requires the explicit computation of a Lyapunov function and the determination by backward numerical integration of a limited number of trajectories. The obtained estimates are close approximations of the exact regions of asymptotic stability of the system’s stable equilibria.