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Julia sets of Newton’s method for a class of complex-exponential function F(z)=P(z)e Q(z)

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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 \(F(z)=ze^{z^{w}}\) (w∈ℂ) through iteration method, and analyze the attracting region of the two fixed points 0 and ∞ 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 ∞ for w=2n (n=0,±2,±4,…) are symmetrical about x-axis and y-axis; select the main argument to be in [−π,π), for arbitrary w=α (α∈ℂ), the attracting regions of the fixed points 0 and ∞ are symmetrical about the x-axis; (3) The attracting regions of the two fixed points 0 and ∞ of J set for w=±η have rotational symmetry of η 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 θ z in the negative x-axis is not continuous, the fault and rupture of the attracting regions of the two fixed points 0 and ∞ appear only in the negative x-axis.

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Authors and Affiliations

  1. School of Electronic & Information Engineering, Dalian University of Technology, Dalian, 116024, China

    Xing-Yuan Wang, Yi-Ke Li, Yuan-Yuan Sun, Jun-Mei Song & Feng-Dan Ge

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  1. Xing-Yuan Wang
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  2. Yi-Ke Li
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  3. Yuan-Yuan Sun
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  4. Jun-Mei Song
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Wang, XY., Li, YK., Sun, YY. et al. Julia sets of Newton’s method for a class of complex-exponential function F(z)=P(z)e Q(z) . Nonlinear Dyn 62, 955–966 (2010). https://doi.org/10.1007/s11071-010-9777-4

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