Numerical determination of the basin of attraction for asymptotically autonomous dynamical systems

Abstract

We develop a method to numerically analyse asymptotically autonomous systems of the form x ̇ = f ( t , x ) , where f ( t , x ) tends to g ( x ) as t → ∞ . The rate of convergence is not limited to exponential, but may be polynomial, logarithmic or any other rate. For these systems, we propose a transformation of the infinite time interval to a finite, compact one, which reflects the rate of convergence of f to g . In the transformed system, the origin is an asymptotically stable equilibrium, which is exponentially stable in x -direction. We consider a Lyapunov function in this transformed system as a solution of a suitable linear first-order partial differential equation and approximate it using Radial Basis Functions.