Finding all periodic orbits of maps using Newton methods: Sizes of basins
type: Item
instance of: scholarly article
title: Finding all periodic orbits of maps using Newton methods: Sizes of basins
zbMATH Open document ID: 0939.37008
DOI: 10.1016/S0167-2789(99)00138-4
author: Yanyan Li
published in: Physica D
publication date: 2000-07-03T00:00:00Z
review text: The paper is devoted to finding \(k\)-periodic orbits of a diffeomorphism \(F\) on \(\mathbb{R}^2\) by applying Newton's method to the function \(G=F^k-I\), where \(I\) is the identity function. The authors actually use some variants of Newton's method (continuous Newton and quasi-Newton methods) that are more robust than the traditional Newton's method. For a \(k\)-periodic point \(p\in \mathbb{R}^2\), the most attention is focused on the size of the Newton basin \[ B(p,k)=\{x\in \mathbb{R}^2:\text{ Newton's method starting from }x \text{ converges to} p\} \] and the manner it depends on \(p\) and \(k\). It is shown that if \(p\) is an attracting point of \(F\) then there is an open neighborhood of \(p\) that is in all Newton basins \(B(p,nk)\) for \(n=1,2,\dots\). If \(p\) is a repelling point then it is possible that \(p\) is the only point which is in all of the Newton basins \(B(p,nk)\) for all \(n\). The main result states that if \(p\) is a saddle point, then the intersection of Newton basins \(B(p,nk)\) includes a segment of the local stable manifold of \(p\). Numerical data presented in the paper indicate that the area of the basin \(B(p,k)\) of a periodic saddle point \(p\) is proportional to \(\lambda ^c\), where \(\lambda\) is the unstable eigenvalue of \(DF^k(p)\) and \(c\approx -1\). From this it follows that for long periods many periodic orbits of \(F\) have \(\lambda\) so large that the basins are numerically undetectable. The results described above are illustrated by detailed numerical analysis of the Hénon and the Ikeda mappings.
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37M05
Mathematics Subject Classification ID: 49M15
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 37E15
Mathematics Subject Classification ID: 37C75
zbMATH DE Number: 1398660
zbMATH Keywords: periodic orbits
zbMATH Keywords: Newton methods
zbMATH Keywords: continuous Newton
zbMATH Keywords: Newton basin
zbMATH Keywords: dynamical system
zbMATH Keywords: diffeomorphism
zbMATH Keywords: periodic saddle point
zbMATH Keywords: Hénon mapping
zbMATH Keywords: Ikeda mapping
reviewed by: Eugene Ershov
describes a project that uses: Dynamics
MaRDI profile type: Publication
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full work available at URL: s0167-2789(99)00138-4
OpenAlex ID: W1983142535
Wikidata QID: Q127373796
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description: scientific article; zbMATH DE number 1398660