Attracting orbits in Newton’s method

Hurley, Mike

doi:10.1090/S0002-9947-1986-0849472-6

Attracting orbits in Newton’s method
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Trans. Amer. Math. Soc. 297 (1986), 143-158 Request permission

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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