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 ∈ R2 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 ∈ C1(R2,R) such that LW(x) =W (x) + L(x) is negative for all x ∈ 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 LW(x) = −μ f (x) . We approximate the function W, which satisfies the linear partial differential equation W (x) = ∇W(x), f (x) =−μ f (x) −L(x), using radial basis functions and obtain an approximation w such that Lw(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. © 2007 Elsevier Inc. All rights reserved.