Determination of the Basin of Attraction using Radial Basis Functions
by
Peter Giesl
University of Sussex, UK

We consider a dynamical system given by a general autonomous ordinary differential equation. The basin of attraction of an equilibrium can be determined by sublevel sets of a Lyapunov function, i.e. a function that is decreasing along solutions. We characterize a Lyapunov function as the solution of a certain linear partial differential equation. In order to calculate the Lyapunov function, we approximate solutions of this partial differential equation using Radial Basis Functions. This approximation uses as ansatz a linear combination of shifted basis functions to interpolate scattered data. If the data points are dense enough, the approximation itself is a Lyapunov function. We present error estimates and discuss local and global properties.

Date received: January 24, 2008