On the determination of the basin of attraction of a periodic orbit in two-dimensional systems

We consider the general nonlinear differential equation x = f (x) with x epsilon R-2 and develop a method to determine the basin of attraction of a periodic orbit. Borg's criterion provides a method to prove existence, uniqueness and exponential stability of a periodic orbit and to determine a subset of its basin of attraction. In order to use the criterion one has to find a function W epsilon C-1 (R-2, R) such that L-W (x) = W'(x) + L(x) is negative for all X epsilon K, where K is a positively invariant set. Here, L(x) is a given function and W'(x) denotes the orbital derivative of W. In this paper we prove the existence and smoothness of a function W such that L (W) (x) = -mu parallel to f (x) parallel to. We approximate the function W, which satisfies the linear partial differential equation W'(x) = = -mu parallel to f (x) parallel to - L(x), using radial basis functions and obtain an approximation omega such that L-omega (x) < 0. Using radial basis functions again, we determine a positively invariant set K so that we can apply Borg's criterion. As an example we apply the method to the Van-der-Pol equation.