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Indexed by:期刊论文

Date of Publication:2010-12-01

Journal:NONLINEAR DYNAMICS

Included Journals:EI、SCIE、Scopus

Volume:62

Issue:4

Page Number:955-966

ISSN No.:0924-090X

Key Words:Newton's method; Julia set; Complex-exponential function; Fixed point; Attracting region

Abstract:In this paper, we analyze the theory of the Julia set (J set) of Newton's method, construct the Julia sets of Newton's method of function F(z) = z(ezw) (w is an element of C) through iteration method, and analyze the attracting region of the two fixed points 0 and infinity when w are different values. Consequently, we draw the following conclusions: (1) When the judge conditions for the iterative algorithm are changed to |N(z(n) ) - z(n) | <= EOF, the properties of the figures in our experiments are contrary to the conclusions in (Wegner and Peterson, Fractal Creations, pp. 168-231, 1991); (2) The attracting regions of the fixed points 0 and infinity for w=2n (n=0, +/- 2, +/- 4, ...) are symmetrical about x-axis and y-axis; select the main argument to be in [-pi,pi), for arbitrary w=alpha (alpha is an element of C), the attracting regions of the fixed points 0 and infinity are symmetrical about the x-axis; (3) The attracting regions of the two fixed points 0 and infinity of J set for w=+/-eta have rotational symmetry of eta times; (4) If w=-4.7, k=0.8, then the attracting regions of different magnifications display a startling similarity, J set holds infinite self-similar structures; (5) When w is a complex number, because the selection of main argument theta(z) in the negative x-axis is not continuous, the fault and rupture of the attracting regions of the two fixed points 0 and a appear only in the negative x-axis.