Chaotic behaviour in the Newton iterative function associated with Kepler's equation
type: Item
instance of: scholarly article
title: Chaotic behaviour in the Newton iterative function associated with Kepler's equation
zbMATH Open document ID: 0981.70003
DOI: 10.1023/A:1008339416143
author: Laura Stumpf
published in: Celestial Mechanics and Dynamical Astronomy
publication date: 2002-03-13T00:00:00Z
review text: This paper is concerned with the behaviour of Newton's method when applied to the solution of Kepler's equation which appears in the two-body problem of celestial mechanics. In this equation for given values of mean anomaly \(M\) and eccentricity \(e\) of elliptic orbit, the eccentric anomaly \(E\) is defined by the implicit equation \( E - e \sin E = M \). As remarked by several authors [see e.g. \textit{E. D. Charles} and \textit{J. B. Tatum}, Celest. Mech. Dyn. Astron. 69, No.~4, 357-372 (1998; Zbl 0947.70003)], for some values of \(e\) close to one and small values of the mean anomaly, Newton's iteration shows a chaotic behaviour. Here the author provides a theoretical explanation of such behaviour by using the so-called Schwarzian derivative together with a theorem of \textit{R. L. Devaney} [A first course in chaotic dynamical systems. Theory and experiment. Studies in Nonlinearity Reading, MA: Addison-Wesley, xi, 302 p. (1992; Zbl 0768.58001)] in which the sign of the derivative of a function its related to its basin of attraction. Further, for a given iteration function and for the starting value of eccentric anomaly \(E= \pi\), some sufficient conditions are derived for the convergence to the mean anomaly and eccentricity.
Mathematics Subject Classification ID: 70-08
Mathematics Subject Classification ID: 70F05
Mathematics Subject Classification ID: 65H05
zbMATH DE Number: 1442307
zbMATH Keywords: Kepler's equation
zbMATH Keywords: Newton's method
zbMATH Keywords: chaotic behaviour
zbMATH Keywords: two-body problem
zbMATH Keywords: mean anomaly
zbMATH Keywords: eccentricity
zbMATH Keywords: eccentric anomaly
zbMATH Keywords: Schwarzian derivative
zbMATH Keywords: basin of attraction
zbMATH Keywords: sufficient conditions for convergence
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: a:1008339416143
OpenAlex ID: W2124211717
Wikidata QID: Q61893622
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description: scientific article; zbMATH DE number 1442307