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

We consider the general nonlinear differential equation 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.