Chaotic behaviour in the Newton iterative function associated with Kepler's equation (Q1976131): Difference between revisions

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.