We study the basin of attraction of an asymptotically stable
equilibrium of a general autonomous ordinary differential equation.
Sublevel sets of Lyapunov functions provide subsets of the
basin of attraction.
In this paper we construct a Lyapunov function by approximation
via radial basis functions. We show the existence and the smoothness
of a Lyapunov function with certain, given orbital derivative.
By approximation of this Lyapunov function via its orbital derivative
using radial basis functions we obtain a global Lyapunov function and
can thus determine each compact subset of the basin of attraction.