An efficient method for computing invariant manifolds of planar maps
type: Item
instance of: scholarly article
title: An efficient method for computing invariant manifolds of planar maps
zbMATH Open document ID: 0768.65025
author: Dana Hobson
published in: Journal of Computational Physics
publication date: 1993-05-16T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65H17
Mathematics Subject Classification ID: 37D99
zbMATH DE Number: 167045
zbMATH Keywords: dynamical systems
zbMATH Keywords: invariant manifolds
zbMATH Keywords: fixed points
zbMATH Keywords: diffeomorphism
zbMATH Keywords: saddle point
zbMATH Keywords: unstable manifold
zbMATH Keywords: linearization
zbMATH Keywords: numerical examples
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: jcph.1993.1002
OpenAlex ID: W2026989737
DOI: 10.1006/JCPH.1993.1002
description: scientific article; zbMATH DE number 167045