M and J sets from Newton's transformation of the transcendental mapping F(z) = e(z(w)+c) with vcps

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

21 Scopus Citations
View graph of relations


Related Research Unit(s)


Original languageEnglish
Pages (from-to)371-383
Journal / PublicationComputers and Graphics (Pergamon)
Issue number2
Publication statusPublished - Apr 2002


Newton's transformation fw(z) = z - 1/(wzw-1) containing only one complex parameter w (w ≠ 0 or 1) is constructed from the transcendental mapping F(z) = e(z(w)+c). Although the number of critical points of fw(z) is countably infinite, a method based on the Valid Critical Point Set, vcps = {zk ∈ C| - π <arg(zk) ≤ π, fw′(zk) = 0, k ∈ Z}, is discussed for the generation of the generalized Mandelbrot set M of fw(z). The petal fragments, the multi-period fragments, and the classical "Mandelbrot set" fragments are found in M. The dynamical characteristics of fw(z) for different values of w are analyzed. The relationship between the parameter w in a classical "Mandelbrot set" fragment and the structure of the corresponding filled-in Julia set is investigated. A large number of novel images of filled-in Julia sets are generated based on the rate of convergence of orbits. © 2002 Published by Elsevier Science Ltd.

Research Area(s)

  • Chaos, Fractal, Julia set, Mandelbrot set, Newton's transformation