Significant Curves of the Mandelbrot Set

MENDEL — Soft Computing Journal, Volume 2d, No.gk, .2+2K#2` 2021, Brno, Czech RepublicX ISSN: 1803-3814 (Printed), 2571-3701 (Online) https://doi.org/10.13164/mendel.2021.k.0jy Significant Curves of the Mandelbrot Set Dalibor Martiˇsek Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic ma[email protected] Abstract The paper provides a description of some interesant curves contained in the Mandelbrot set. These curves are the boundaries of the areas called bulbs“ which are described approximately only in present. In this paper, some of them are described analyticaly – curves of so called first period, the boundary of the main hyperbolic component, internal and external bounds and also some curves of the second period. Keywords: Mandelbrot Set, Main Hyperbolic Component, Internal Bound, External Bound, Resultant of Polynomials. Received: 02 October 2021 Accepted: 03 December 2021 Published: 21 December 2021 1 Introduction The Mandelbrot set was discovered in 1979 during an attempt to castalogize the sets that Gaston Julia and Piere Fatau studied in the second decade of the 20th century. It is a dynamic system whose behavior is described by the equation z0=0; zn+1 =z2 n+c;n∈N;zn;c∈C(1) or more generally z0=0; zn+1 =f(zn;c); n∈N;zn;c∈C(2) where f(z) is a holomorfic function. This series converges for some values of c and diverges for others points. The Mandelbrot set consists of all points c∈C, for which the process (1) or (2) does not diverge. Formally: M(f)=c∈C|lim n→∞ f(zn;c)=∞;n∈N;zn;c∈C (3) 2 Visual Representation of the Mandelbrot Set The visual representation of the Mandelbrot set may be created by determining, for each point C∈Cof a part of the complex plane, whether znis bounded. The number of iterations to reach a chosen radius can be used to determine the color to use – it is so called the Integer Escape-Time (IET) algorithm [3,1,5]. However, this algorithm creates clearly visible colour discontinuities – see the upper part of Fig. 1. The socalled Smooth Escape-Time algorithm is more fitting for visualizations. It is applicable to polynomial function f(z)=zk+ck−1zk−1+...+c1z+c0≈zk+c0(because |z|1 therefore |zk||zk−1|). Let us assume that r is chosen escape radius and the divergence is detected in the n-th step. In this case, |zn|∈r;rkand (constant) n-th colour belongs to this whole interval in IET algorithm. It is necessary to calculate a parameter h∈(0; 1) to assign “n+h”-th colour for continuous transition. We need a suitable function for the transformation r;rk→(0; 1). Value h=log klogr|zn|is applicable for this purpose for example – see the lower part of Fig. 1. In the following text, we will analyse sets Mk(f) – sets according to (3) for which f(zn;c)=zn+1 =zk n+c. (4) 3 Curves of the first period Curves of the first period are sets C1=z∈Cz=zk+c. 3.1 The Boundary of the Main Hyperbolic Component The main hyperbolic component H(1) kis the subset of Mk(f(z)) for which the orbit does not diverge [2,4]. It is bounded by C1. In case of (4), f(z)=zk+c(5) and equation of C1is possible to obtain from f(z)= z=zk+c, i.e., c=z−zk.(6) According to Banach fixed-point theorem, its points must satisfy the condition |f(z)|≤1 and for its boundary |f(z)|=1. Itmeansthatf(z)=eiω in the Gaussian plane. Therefore, if we assign z=k−1 1 kexp(iω)=1 kexp(iω)1 k−1 =1 k1 k−1 exp iω k−1(7) 30 MENDEL — Soft Computing Journal, Volume 2d, No.gk, .2+2K#2` 2021, Brno, Czech RepublicX Figure 1: Visualization of the Mandelbrot set. Integer Escape-Time (up), Smooth Escape-Time (down). then f(z) (5)  =(zk+C)=k·zk−1 =k·1 k1 k−1 exp iω k−1k−1 =exp(iω) (8) and c (6)  =z−zk (7)  =1 k1 k−1 exp iω k−1−1 kk k−1 exp iω k−1 (9) For simplification, let us denote iω k−1=ϕand we obtain boundaries of the main hyperbolic components as c(1) k=1 k1 k−1·exp (iϕ)−1 kk k−1·exp (iϕk); 0 ≤ϕ<π (10) The subscript (1) means that this is the curve of period one – if znlies on this curve, then also zn+1 lies on the curve too. For the classic Mandelbrot set zn+1 =z2 n+C is k= 2 and we have c(1) 2(ϕ)=1 2eiϕ −1 4e2iϕ =1 2(cos ϕ+i·sin ϕ)−1 4(cos 2ϕ+i·sin 2ϕ) For zn+1 =z3 n+Cis k= 3 and c(1) 3(ϕ)= 1 √3eiϕ −1 3e3iϕ =1 √3(cos ϕ+i·sin ϕ)−√3 9(cos 3ϕ+i·sin 3ϕ) By analogy c(1) 4(ϕ)= 1 3 √4eiϕ −1 4e4iϕ =1 3 √4(cos ϕ+i·sin ϕ)−1 43 √4(cos 4ϕ+i·sin 4ϕ) These curves are marked as pink on Figures 2, 3, and 4. It is important to note that limk→∞ 1 k 1 k−1 = 1 and limk→∞ 1 k k k−1 = 0 in (10) and therefore lim k→∞ c(1) k(ϕ)=eiϕ, which means that these curves converge into the unit circle. 31 MENDEL — Soft Computing Journal, Volume 2d, No.gk, .2+2K#2` 2021, Brno, Czech RepublicX J`iBb2F,gaB;MB7B+Mig*m`p2bgQ7gi?2gJM/2H#`Qiga2i Figure 2: Curves of the first and second period for k=2. 3.2 Internal Bound For speeding up rendering of the Mandelbrot set, it is possible to detect as internal points those belonging into the curve of period one. They lie on the circle with the centre S= 0 and radius r=minc(1) k. Because c(1) k 2=1 k1 k−1 cos ϕ−1 kk k−1 cos kϕ2 +1 k1 k−1 sin ϕ−1 kk k−1 sin kϕ2 =1 k2 k−1 cos2ϕ−21 k1 k−1 cos ϕ1 kk k−1 cos kϕ +1 k2k k−1 cos2kϕ +1 k2 k−1 sin2ϕ −21 k1 k−1 sin ϕ1 kk k−1 sin kϕ +1 k2k k−1 sin2kϕ =1 k2 k−1 −21 k1 k−1 1 kk k−1 (cos ϕcos kϕ +sinϕsin kϕ) +1 k2k k−1 c(1) k 2=1 k2 k−1−21 k1 k−1 1 kk k−1 cos [(k−1) ϕ]+ 1 k2k k−1 for which the minimum occurs for cos [(k−1) ϕ]=1 (i.e. ϕ= 0), therefore min c(1) k 2=1 k1 k−12 −21 k1 k−1 1 kk k−1 =k1 1−k−kk 1−k2 and r=minc(1) k=k1 1−k−kk 1−k(11) They are circles with the center S= 0 and radius (14), its equations for k= 2; 3; 4 are c(in) 2(μ)=1 4eiμ;c(in) 3(μ)=2√3 9eiμ;c(in) 4(μ)=33 √4 16 eiμ These circles are marked as pink in Figures 2, 3, 4. 3.3 External Bound Rendering of the Mandelbrot set is possible to accelerate also by external bound detection – the escape 32 MENDEL — Soft Computing Journal, Volume 2d, No.gk, .2+2K#2` 2021, Brno, Czech RepublicX Figure 3: Curves of the first and second period for k=3. radius mentioned in section 2 does not have to be bigger than radius of the external bound. We can assume that outside the escape zone |zn|is already bigger than |c|therefore |zn+1| |zn|=zk n+c |zn|>1 Using the anti-triangle inequality |a+b|≥||a|−|b|| for complex numbers we get zk n+c |zn|≥|zn|k−|c| |zn|=|zn|k−1−|c| |zn| Since |zn|>|c|,wehave zk n+c |zn|≥|zn|k−1−|c| |zn|≥|zn|k−1−1≥1⇒ |zn|k−1≥2⇒|zn|≥21 k−1 Which means that the external bounds are the circles with centre S= 0 and radius r=2 1 k−1, their equations for k= 2; 3; 4 are c(out) 2()=2ei;c(out) 3()=√2ei;c(out) 4()= 3 √2ei These circles are marked as pink in Figures 2, 3, 4. 4 Curves of the Second Period Curves of the second period are sets C2=z∈Cz=zk+ck+c−C1 These curves are boundaries between convergence and divergence process z=zk+ck+c According to Banach fixed-point theorem, the following relationship |f(z)|=(zk+ck+c)= 1 (12) must hold again. 4.1 Curves of the second period for k=2 We have to find roots of z=z2+c2+c⇒z4+2z2c−z+c(c+1)=0 that are not roots of z=z2+c⇒z2−z+c=0 33 MENDEL — Soft Computing Journal, Volume 2d, No.gk, .2+2K#2` 2021, Brno, Czech RepublicX J`iBb2F,gaB;MB7B+Mig*m`p2bgQ7gi?2gJM/2H#`Qiga2i Figure 4: Curves of the first and second period for k=4. Quotient of these polynomials must be equal to zero z4+2z2c−z+c(c+1) :z2−z+c =z2+z+c+1=0 (13) Equation (12) gives ((z2+c)2+c)=2z2+c·2z=4z3+4cz=1 Therefore 4z3+4cz =eωi Using 1 4eωi =λ(14) we obtain z3+cz −1 4λ=0⇒z3+cz −λ= 0 (15) Now, it is necessary to calculate the resultant of polynomials (13) (15), i.e., the expression of their coefficients, which is equal to zero if and only if the polynomials have a common root. Therefore, we must construct the determinant of the Sylvester matrix and set it equal to zero.  111+c00 01 1 1+c0 00 1 1 1+c 10 c−λ0 01 0 c−λ  =(1+c−λ)2=0 We obtain 1 + c−λ= 0 and from substitution (14) c(2) 2(ω)=1 4eωi −1 This circle is marked as black in Figures 2, 3, 4. 4.2 Curves of the second period for k=3and 4 According to previous section: for k=3,wehaveto find roots of z=z3+c3+c⇒z9+3cz6+3c2z3−z+c3+c=0 that are not roots of z=z3+c⇒z3−z+c=0 34 MENDEL — Soft Computing Journal, Volume 2d, No.gk, .2+2K#2` 2021, Brno, Czech RepublicX Quotient z9+3cz6+3c2z3−z+c3+c:z3−z+c =z6+z4+2cz3+z2+cz +c2+1=0 (16) Equation (12) gives z3+c3+c=3z3+c2·3z2 =9z2z3+c2=1 Therefore 9z2z3+c2=eωi Using 1 9eωi =λ(17) we obtain z8+2z5c+c2z2−λ= 0 (18) Determinant of Sylvester matrix of resultant of (16)- (18) is c2−(λ−1) (λ+1) 22=0 and for c,wehave c(2) 3(ω)=1 3λ(λ2+3λ−9) −1 =1 27eωi (e2ωi +3eωi −9) −1 For k= 4, we can obtain c(2) 4(ω)= 3 1 44(e4ωi +4e3ωi +16e2ωi −64eωi)−1 The calculation is already technically difficult, so we do not perform it. 5 Conclusion In [4], we can read about the Mandelbrot set of the second degree: “It should be pointed out that the bulbs’ apparent circular shape is indeed only approximate.” In this paper, we proved, that the curve of the second period in this set is precise circle. The curves of the higher period cannot be precise circles because the Mandelbrot set is generated by nonlinear transforms. However, it is possible to obtain their analytical description on principle. Acknowledgement: The author acknowledges support from Private Institute of Applied Mathematics, Slapanice, Czech Republic. References [1] Devaney, R. The fractal geometry of the mandelbrot set ii: How to add and how to count. Fractals 03, 04 (1995), 629–650. [2] Devaney, R. Unveiling the mandelbrot set. Plus Magazine 40 (2006). [3] Douady, A., and Hubbard, J. H. It´eration des polynˆomes quadratiques complexes (iteration of complex quadratic polynomials). C. R. Acad. Sci., Paris, S´er. I 294 (1982), 123–125. [4] Fowler, A., and McGuinness, M. Thesizeof mandelbrot bulbs. Chaos, Solitons & Fractals: X 3(2019), 100019. [5] Goldberg, L., and Tresser, C. Rotation orbits and the farey tree. Ergodic Theory and Dynamical Systems 16, 5 (1996), 1011–1029. 35