An efficient method for computing invariant manifolds of planar maps
A method to compute invariant manifolds of fixed points of planar maps is proposed. Assuming that a map \(f:\mathbb{R}^ 2\to\mathbb{R}^ 2\) is a \(C^ r\)- diffeomorphism (\(r \geq 1\)) and it has an orientation preserving saddle point \(x^*\), the problem is to determine numerically either the stable or unstable manifolds of \(x^*\). The usual approach to compute the unstable manifold starts obtaining by linearization an ordered set of points \(p_ 0,p_ 1,\dots,p_ k = f(p_ 0)\) along the unstable manifold and very close to \(x^*\). Then, taking further iterates of these points, one gets a discrete set of points on the desired manifold. However, since the distance between the points of successive iterates increases rapidly with the number of iterations, the shape of the manifold can not be accurately determined when we advance along the curve unless the points taken initially between \(p_ 0\) and \(f(p_ 0)\) is very large. The method proposed here attempts to overcome this difficulty by introducing intermediate points \(u_ j\) between two consecutive iterates \(p\) and \(f(p)\) so that the arclength between \(u_ j\) and \(u_{j+1}\) is approximately constant. This is done in practice bounding appropriately some estimate of the local radius of curvature. Finally, some practical suggestions on the method are given and several numerical examples are presented to show the reliability of the new method.
scientific article; zbMATH DE number 167045