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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Attracting orbits in Newton's method


Author: Mike Hurley
Journal: Trans. Amer. Math. Soc. 297 (1986), 143-158
MSC: Primary 58F12; Secondary 26C10
DOI: https://doi.org/10.1090/S0002-9947-1986-0849472-6
MathSciNet review: 849472
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Abstract: It is well known that the dynamical system generated by Newton's Method applied to a real polynomial with all of its roots real has no periodic attractors other than the fixed points at the roots of the polynomial. This paper studies the effect on Newton's Method of roots of a polynomial "going complex". More generally, we consider Newton's Method for smooth real-valued functions of the form $ {f_\mu }(x) = g(x) + \mu $, $ \mu $ a parameter. If $ {\mu _0}$ is a point of discontinuity of the map $ \mu \to $ (the number of roots of $ {f_\mu }$), then, in the presence of certain nondegeneracy conditions, we show that there are values of $ \mu $ near $ {\mu _0}$ for which the Newton function of $ {f_\mu }$ has nontrivial periodic attractors.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0849472-6
Article copyright: © Copyright 1986 American Mathematical Society