TY - JOUR
T1 - M and J sets from Newton's transformation of the transcendental mapping F(z) = e(z(w)+c) with vcps
AU - Chen, Ning
AU - Zhu, X. L.
AU - Chung, K. W.
PY - 2002/4
Y1 - 2002/4
N2 - 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 k) ≤ 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.
AB - 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 k) ≤ 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.
KW - Chaos
KW - Fractal
KW - Julia set
KW - Mandelbrot set
KW - Newton's transformation
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U2 - 10.1016/S0097-8493(01)00185-6
DO - 10.1016/S0097-8493(01)00185-6
M3 - 21_Publication in refereed journal
VL - 26
SP - 371
EP - 383
JO - Computers and Graphics (Pergamon)
JF - Computers and Graphics (Pergamon)
SN - 0097-8493
IS - 2
ER -