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Journal of Global Scientific Research (ISSN: 2523-9376) Global Scientific 2 (2020) 356-361 Research

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New Julia Set by Iteration of Rational

Zainab Weli Murad

Department of , College of Science, University of Kirkuk, Kirkuk, Iraq

Email: [email protected]

Received: 5 March 2020, Revised: 1 April 2020, Accepted: 25 May 2020, Online: 11 June 2020

Abstract This paper studies the properties of a Julia set that appear when it is constructed via the relaxed Newton's method on 훿 the function 푔(푧) = 푧휇푒푧 , where µ and δ are real numbers, have the same value but opposite in signal. Julia set was the of all brightly colored attracting regions for the fixed point (0) at z-plane. Moreover, the boundary of these attracting regions has a fractal structure. All images have been generated by a computer program by n n considering the condition |N l(zn) | ≤ 0.001. Furthermore, by a condition {Nl (z0)}, we determined the color of attracting regions where the black indicated to a divergent region and the colors indicated to attracting regions. Also, we used different iterative degree for every structure to obtain various colors. So when δ ≤ −3 together with μ ≥ 3, the images looked like shapes in v- plane. Finally, we inserted some images of arbitrary μ and δ using a different iterative number (n ≤ 50) to determine more variety of pretty colors. The results are verified and approved using computer Figures.

Keywords: Julia set, Relaxed Newton’s method, Fractal, computer Figures.

1. Introduction analytic complex map divides the complex plane into two split subsets: 1) The stable set, which called Fatuo Newton’s method has many procs and cons in set(퟊); and 2) Julia set(τ). These sets borders can compare with Relaxed Newton’s method. For example, create nice figures that defined by . Fractal - in Newton’s method, the multiple roots cannot be likes structures whose beauty and complexity is only determined before the solution of the equation. The rivalled by Nature itself. Furthermore, Fractals are the Relaxed Newton's method, which applies to the resultant of rational functions iterations, and the of the form, R(z) = P(z) ⁄ Q(z) is performance of rational functions which have high R(z) iterates is crucial in the dynamic of rational maps. defined by N (z) = z − (l ∗ ). The root of R(z) l R′(z) Back in the 1970s and 1980s, mathematicians working corresponds to attracting fixed points of Nl(z). in an area called dynamical systems made use of the Besides, because of the exponential term, the relaxed ever-advancing computing power to draw computer Newton iteration function has a rationally indifferent images of the objects they were working on. Up until δ fixed point at infinity if it's applied on g(z) = zμez . now, the graphing algorithm for the fractal via the Furthermore, the Fatuo Flower theorem can give an computer is a fruitful direction for research [1]-[6]. analytic description of the dynamics around a However, using the relaxed Newton's method to solve rationally indifferent fixed point. Therefore, the equations in z- plane presents interesting images refer ∗ degree of the exponent of Q(z) = zδ completely to attracting region Α(z ). The region defined to be determines the number of petals at infinity. An the set of all points that converge to the same root. The attracting region for a rationally indifferent fixed

Murad, Z. W. Journal of Global Scientific Research ISSN: 2523-9376 Vol. 2 357

푁′(푧) = (1 − 푙) + 푙 ∗ 푅(푧)푅″(푧)/((푅′(푧))2 point lied in fatuo set ((퟊Nl), ) and the border of these 푙 attracting regions are Julia set (τ ) . Furthermore, Nl (3) each repelling points having a place with Julia set(τN ). Moreover, all the attracting fixed points of l If R′(z∗) ≠ 0, and the order of root (l ≥ 2), thereafter rational functions having a place with Fatou set ′ ∗ ∗ |Nl (z )| ≥ 1. Therefore, it means that z is repelling ((퟊Nl), )[7]–[9]. fixed point of Nl(z). In the same way, if l ∈ [0,1], then |N′(z∗)| ≤ 1 and z∗ is attracting fixed point of The aim of this article is studies the properties of a l N′(z) . The Relaxed Newton's approach is usually Julia set that appear when it is constructed via the l implemented in solving the complex equations that relaxed Newton's method on the function 푔(푧) = create pretty pictures. It is characterized by attracting 휇 푧훿 푧 푒 , where µ and δ are real numbers, have the same regions that contain every point converge to the same value but opposite in signal. Julia set was the root of R(z) = 0, while n converging to infinity [10]. boundary of all brightly colored attracting regions for the fixed point (0) at z-plane. So when δ ≤ −3 together 3. Julia Set of Relaxed Newton's with μ ≥ 3, the images looked like shapes in v- plane Method furthermore this study is an extension of δ previous work in [10] of the authors (X.-Y. Wang, Y.- The attracting regions of g(z) = zμez is an open K. Li, Y.-Y. Sun, J.-M. Song, and F.-D. Ge) , which district including z∗. Additionally, z∗ is considered as they analyzing the structure characteristics of the a super-attracting fixed point of Nl(z) and every attracting region of Julia sets of the relaxed newton's attracting region is corresponding to various colors. Q(z) ( ) The Julia set (τN ) of Nl(z) is known as the method (τNl) of g z = p(z)e . Here, we change l p(z) = z to general p(z) = boundaries between the attracting regions, every point zμ where µ ≥ 1 and Q(z) = zδ, where δ ∈ ℂ, and δ in attracting regions boundary and in each other ≤ 1. Finally, the outcomes of this study are proved via attracting region’s boundary as well. However, these computer images. points are not convergent for any relaxed Newton's method and shape fractals [10]. 2. Classical and Relaxed Newton's 4. The Method For Construct Julia Set Method Dynamics On Rational Functions Fractals Via Computer Programming Let us define the Rational function by R = P ⁄ Q , μ zδ where P and Q are complex with no To construct Julia set of Nl(z) on the g(z) = z e , common factors, and the degree of R is defined where δand μ are a real number, have the same value ∗ by d = deg(R) = max{deg(P), deg(Q)}. So for, a but opposite in signal. We must find the root zi P : (i = 1,2,3, … , d) of g(z) = 0, where d is the order of complex polynomial ℂ → ℂ of degree (d ≥ 2), a R(z). Additionally, modify the iterative model to complex Newton's Transformation ( N(z): ℂ →ℂ ) ∞ ∞ zn+1 = Nl(z) and put zn( n = 0,1,2,3, … … M) . If of R(z) is defined by: n |N l(zn) | ≤ 0.001 then we can consider that zn+1 is the root of g(z) = 0 . where n is the number of 푅(푧) 푁(푧) = 푧 − ( ) (1) iterations. Furthermore, we assign the color of the 푅′(푧) point z0 depending on the convergence condition of n orbit {N l(z0)}. More specifically, the black denotes a Fixed points of N(z) are roots of the R(z) and N(z)= z. divergent region, and other colors refer to attracting Newton's algorithm convergent quadratically to a region. In these fractals, the boundary of all brightly simple root, but its converges linearly for multiple colored attracting regions is the Julia set, which roots of R(z) = 0. Therefore, the relaxed Newton's structures by this method. Besides, we analyze Julia method has advantages and limitations in comparison set of Nl(z) according to Theorem 1 and 2, while the with Newton’s method. The relaxed Newton's amount of l = 1 diminishes and the attracting region transformation map (Nl(z): ℂ∞→ℂ∞) of R(z) on the in the starting point is growing. Finally, using the can be defined by: different iterative number, we can achieve variety and more pretty colors [10-15].

푅(푧) 푁 (푧) = 푧 − (푙 ∗ ) (2) Theorem 1: When constructing Julia set of Nl(z), for 푙 푅′(푧) δ a function g(z) = zez with the iteration method, if δ = ∓ n , then Julia set holds n times of rotational If z∗ is a simple root of R(z), then z∗ is fixed points of [10]. NR,l(z) satisfying NR,l(z) = z. Additionally, it is Theorem 2: When constructing Julia set of Nl(z) for δ possible to limit the type of fix points via the a function g(z) = zez with iteration method, if δ = 2n following equation: (n = 0, ∓2,∓4,...), thereafter Julia set is symmetrical around x and y axes, If the major argument is within Murad, Z. W. Journal of Global Scientific Research ISSN: 2523-9376 Vol. 2 358

[−π,π), thereafter for any value (δ is real number), Julia set is symmetrical about x-axis [10].

5. Experiment and Results

δ Via Relaxed Newton's method on (z) = zμez , where P(z) = zμ , Q(z) = zδ ,µ and δ are real numbers, have the same value but opposite in signal.Then, we can get the following equations:

푔(푧) 푧휇 푁 (푧) = 푧 − 푙 = 푧 − 푙 (4) 푙 푔′(푧) (휇푧휇−1+푧휇훿푧훿−1)

푰풎풂품풆 ퟑ: µ = ퟏ, 휹 = −ퟏ, 풍 = ퟎ. ퟔ 훿 2 ′ −휇+푧 (훿 −훿) 1 푁푙 (푧) = 1 − 푙 2 . 2 (5) 푧 휇+훿푧훿 ( ) 푧 Julia set with δ = − 1, μ =1 and l = 1, is like twain buds at opposite portions from the origin which have no symmetry. Additionally, when δ = −2 with μ = 1 Because of g(z) = 0 , 0 is a fixed point of N (z) . l and l = 0.05, Julia set is like twain buds joint together Additionally, N′(z) = 1 − l if 0 < l < 1 and the l with an x-axis with n little petals. Accordingly, as attracting point of N (z) is zero. In this article, we l shown in Image 1 and 4, it is symmetrical around the used computer programs to generate images by x and y axes. n ( ) adopting the condition |N l zn | ≤ 0.001.

푰풎풂품풆 ퟏ: µ = ퟏ, 휹 = −ퟏ, 풍 = ퟏ

푰풎풂품풆 4: µ = 2, 훿 = −2, 푙 = 1

푰풎풂품풆 ퟐ: µ = ퟏ, 휹 = −ퟏ, 풍 = ퟎ. ퟖ

퐼푚푎푔푒 5: µ = 2, 훿 = −2, 푙 = 0.05

Murad, Z. W. Journal of Global Scientific Research ISSN: 2523-9376 Vol. 2 359

푰풎풂품풆 ퟔ: 흁 = ퟐ, 휹 = −ퟐ, 풍 = ퟎ. ퟏ 푰풎풂품풆 ퟗ: 흁 = ퟑ, 휹 = −ퟑ, 풍 = ퟎ. ퟏ At the point, when μ ≥ 3 and δ ≤ −3 with different values of l, Julia set looks like a flower for n standard petals because it holds rotational symmetry. Each principle petals consists of n small petals, and each small petal consists of n fewer petals. Thereby, the small petals have a similarity and fractal character of the main petals (see the following images).

푰풎풂품풆 ퟏퟎ: µ = ퟒ, 휹 = −ퟒ, 풍 = ퟏ

푰풎풂품풆 ퟕ: µ = ퟑ, 휹 = −ퟑ, 풍 = ퟏ

푰풎풂품풆 ퟏퟏ: µ = ퟒ, 휹 = −ퟒ, 풍 = ퟎ. ퟏ

푰풎풂품풆 ퟖ: 흁 = ퟑ, 휹 = −ퟑ, 풍 = ퟎ. ퟓ

Murad, Z. W. Journal of Global Scientific Research ISSN: 2523-9376 Vol. 2 360

푰풎풂품풆 ퟏퟕ: µ = ퟕ, 휹 = −ퟕ, 풍 = ퟎ. ퟏ 푰풎풂품풆 ퟏퟐ: µ = ퟒ, 휹 = −ퟒ, 풍 = ퟎ. ퟓ

푰풎풂품풆 ퟏퟓ: µ = ퟔ, 휹 = −ퟔ, 풍 = ퟎ. ퟓ 푰풎풂품풆 ퟏퟖ: µ = ퟕ, 휹 = −ퟕ, 풍 = ퟎ. ퟓ From the above images, we can see that when δ and μ together are odd, Julia set hold n times of rotational symmetry. On the other hand, when δ and μ together are even, Julia sets are symmetry around x and y-axes. Finally, the following images are some Julia set fractals for arbitrary μ, δ and l:

푰풎풂품풆 ퟏퟔ: µ = ퟕ, 휹 = −ퟕ, 풍 = ퟏ

푰풎풂품풆 ퟏퟗ: µ = ퟏퟏ, 휹 = −ퟕ, 풍 = ퟎ. ퟐ Murad, Z. W. Journal of Global Scientific Research ISSN: 2523-9376 Vol. 2 361

various colors. When δ ≤ −3 together with μ ≥ 3, the images looked like shapes in v- plane. Finally, we inserted some images of arbitrary μ and δ using a different iterative number (푛 ≤ 50) to determine more variety of pretty colors.

7. References

[1] R. A. Alexander D. A., Iavernaro F., “A History of in One Variable,” Am. Math. Soc., vol. 50, no. 3, pp. 503–511, 2013. [2] D. Alexander, R. L.Devaney, (2013), “A Century of Complex Dynamics”, Simons Foundation, USA, pp. 1-28. [3] R. L.Devaney, (2005), “A First Course in Chaotic 푰풎풂품풆 ퟐퟎ: µ = ퟑ, 휹 = −ퟔ, 풍 = ퟏ Dynamical Systems”, Perseus Books, USA, pp. 1- 293. [4] B. B. Mandelbrot, (1984), "The Fractal Geometry of Nature", The American Mathematical Monthly, vol. 91, no. 9, pp. 594-598. [5] B. B. Mandelbrot, (1989), "Fractal Geometry: What Is It , and What Does It Do?", Proceedings of the Royal Society of London, vol. 423, pp. 2- 16. [6] A. F. Beardon, (1991), "Iteration of Rational Functions", Complex Analytic Dynamical System, Springer – Verlag, USA, pp. 1-272. [7] F.Cilinger,¸ (2007), "Mystery of the Rational Iteration Arising from Relaxed Newton’s Method", Chaos Solutions and Fractals, vol. 32, pp. 471– 479. [8] F.Cilinger, (2004), “Finiteness of the Area of Basins of Attraction of Relaxed Newton Method

for Certain Holomorphic Functions”, 푰풎풂품풆 ퟐퟏ: µ = ퟒ, 휹 = −ퟐ, 풍 = ퟏ International Journal of Bifurcation and Chaos, vol. 14, no. 12, pp. 4177–4190. 6. Conclusion [9] F. Cilinger, (2008), "Infinite Basins of Julia Sets", International Journal of Bifurcation and Chaos, In this article, we studied Julia set constructed by a vol. 18, no. 10, pp. 3169–3173. δ [10] X.-Y. Wang, Y.-K. Li, Y.-Y. Sun, J.-M. Song, relaxed Newton’s method of g(z) = zμez with an and F.-D. Ge, “Julia sets of Newton’s method for iterative method where δ distinctive negative values a class of complex-exponential function F (z)= P with the arbitrary positive values of μ. All images (z) e Q (z),” Nonlinear Dyn., vol. 62, no. 4, pp. have been generated by a computer program by n 955–966, 2010. considering the condition | N l(zn) | ≤ 0.001. n [11] X.-Y. Wang, Yu X. J. (2009), "Julia Set of the Furthermore, by a condition {Nl (z0)}, we determined Newton Transformation for Solving Some the color of attracting regions where the black Complex Exponential Equation”, Fractal, vol. 17, indicated to a divergent region and the colors no. 2, pp. 197–204. indicated to attracting regions. Here, Julia set was the [12] X.-Y. Wang, W. J. Song, X. L. Zou, (2009)," boundary of all brightly colored attracting regions. Julia Set of the Newton Method for Solving Some The attracting regions of a Julia set found to be Complex Exponential Equation ", International symmetry around x and y axes when δ = 2n (n = 0, ±2, Journal Image Graph, vol. 9, no. 2, pp.153–169. ±4...). Moreover, the attracting regions of a Julia set [13] M. E. Haruta , (1999), “Newton's Method on the have had n times rotational symmetry when δ= ±n. Complex Exponential Function”, Transactıons of Besides, when δ ≤ −3 and μ ≥ 3, Julia set obtained as the American Mathematical Society, vol. 351, no. a flower for n principle petals. Each principle petals 6, pp. 2499-2513. consisted of n small petals, and each small petal [14] Z. W. Murad, “Finite Basin’s Area Fractal via contained n fewer petals. Therefore, the little petals Complex Newton’s Method,” kirkuk Univ. J. Sci. have had a similarity and fractal character of the main Stud., vol. 13, no. 1, pp. 262–272, 2018. petals. In addition to the above, by doing large [15] Z. weli Murad, “Infinite Basin’s Area Fractal via magnification, we observed a boundary of attracting Complex Newton’s Method,” Int. J. Appl. Sci. regions have a fractal structure. Also, we used Math., vol. 3, no. 5, pp. 157–161, 2016. different iterative degree for every structure to obtain