Toward the calculation of higher-dimensional stable manifolds and stable sets for noninvertible and piecewise-smooth maps
type: Item
instance of: scholarly article
title: Toward the calculation of higher-dimensional stable manifolds and stable sets for noninvertible and piecewise-smooth maps
zbMATH Open document ID: 1167.37039
author: Danny Fundinger
published in: Journal of Nonlinear Science
publication date: 2009-05-08T00:00:00Z
review text: Stable manifolds can be considered as fundamental building blocks of dynamical systems. Together with its sibling, the unstable manifold, each stable manifold might provide important information about a system's global dynamics. The stable and unstable manifold may indicate, for example, boundaries of basins of attraction or, if a stable and unstable manifold intersect, the occurrence of chaotic dynamics. In this work the author presents a method to compute the stable manifolds and sets of maps in general. Neither the system's inverse nor its Jacobian is required. It is also not limited to one-dimensional stable manifolds and sets. Theoretically, the method might be capable of computing manifolds independent of their dimension, though in practice it applied it only to get one- and two-dimensional stable manifolds and sets. For a given map \(f:\mathbb{R}^d \to \mathbb{R}^d\), the proposed method is capable of yielding large parts of stable manifolds and sets within a certain compact region \(M \subset \mathbb{R}^d\). The algorithm divides the region \(M\) in sets and uses an adaptive subdivision technique to approximate an outer covering of the manifolds. The use of suggested method is illustrated by computation of one- and two-dimensional stable manifolds and global stable sets.
reviewed by: Irina V. Konopleva
Mathematics Subject Classification ID: 37M20
Mathematics Subject Classification ID: 37M99
Mathematics Subject Classification ID: 37-04
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 37E99
zbMATH DE Number: 5552195
zbMATH Keywords: stable manifold
zbMATH Keywords: global stable set
zbMATH Keywords: algorithms
zbMATH Keywords: nonlinear dynamical systems
zbMATH Keywords: noninvertible map
zbMATH Keywords: adaptive subdivision
describes a project that uses: AnT 4.669
MaRDI profile type: Publication
full work available at URL: s00332-007-9016-4
OpenAlex ID: W2071197979
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DOI: 10.1007/S00332-007-9016-4
description: scientific article; zbMATH DE number 5552195