Circular Saw Mandelbrot Set

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1 Circular Saw Mandelbrot Set MAMTA RANI 1, MANISH KUMAR 2 1 Faculty of Computer Systems & Software Engineering University Malaysia Pahang Lebuhraya Tun Razak, Gambang, Kuantan, Pahang Darul Makmur MALAYSIA mamtarsingh@rediffmail.com 2 Department of Computer Science & Engineering Bansal Institute of Engineering & Technology Sitapur Road, Lucknow , Uttar Prdesh INDIA dr.manish.2000@gmail.com Abstract: - Superior Mandelbrot sets, a new class of Mandelbrot sets, have recently been studied for the first time by Rani and Kumar. Indeed, Mandelbrot sets for complex-valued polynomials, under certain circumstances, are particular cases of the corresponding superior Mandelbrot sets. The objective of this paper is to show that the superior Mandelbrot sets for complex-valued polynomials of the form z m +c become circular saw when m becomes large. Further, we show graphically that a sequence of general superior Mandelbrot sets approaches a superior Mandelbrot set. Key-Words: - Superior Mandelbrot set, General superior Mandelbrot set, General superior escape criterion, Circular saw Mandelbrot set. 1 Introduction Mandelbrot sets are very popular infinitely complex fractals. For the last almost three decades, the Mandelbrot set, due to its exciting graphical representations and fascinating properties, has been studied extensively in fractal graphics and complex dynamics. Mandelbrot sets have been used for generating colorful images, melodious music, recently in cryptography [13] etc. For various properties of Mandelbrot set and related studies, one may refer, for instance, to Branner and Hubbard [4], Crilly et al. [5], Crownover [6], Devaney [7] and [8], Kigami [11], Lei [14], Mandelbrot [15], Petigen et al. [23] and [24]. Generally, Mandelbrot set lives in complex plane. D. Rochon [27] studied a more generalized form of Mandelbrot set, which lives in bi-complex plane (see also [9]). Naschie [17] studied the entropy on complex fractals. Mandelbrot set has been analyzed from different aspects, e.g., Argyris et al. studied external and internal perturbation of Mandelbrot set [1] (also see [2] and [30]), Ashlock [3] presented a collection of fitness functions that permit three-parameter evolutionary search of the Mandelbrot set to locate interesting views and Pastor et al. have given an excellent analysis of periodic and chaotic regions in Mandelbrot set (see [20], [21] and [22]). Pastor et al. [28] also studied the nature of the tip and gave two simple formulae to calculate the pattern of the cusp and the tip of a midget in the Mandelbrot set antenna. Joshi et al. [10] proposed an octonionic generalization of the Mandelbrot set, which is sensitive to these transition points and performed an investigation into the limiting dynamics and bifurcation phenomena of the non-associative octonionic quadratic map [12]. In 2003, Rani introduced superior iterates (essentially due to Mann [16]) in the study of fractal theory [25] and created superior Mandelbrot sets [26]. Rani and Kumar [26] showed that for a complex map many superior Mandelbrot sets exist and Mandelbrot set is only a special case. Later on, Negi and Rani [18] studied the effect of dynamic noise on superior Mandelbrot set. Also, they gave analysis of midgets of the same [19]. In this Paper, we introduce the superior iterations on a more general setting in the study of Mandelbrot sets of quadratics, cubics and other complex-valued polynomials and, discuss some related properties. Rani and Kumar [26] conjectured that the superior Mandelbrot set for z m + c is circular when m is large. We have attempted to resolve this issue to some extent, and shown that the superior Mandelbrot sets for z m + c look like a circular saw in many cases. ISSN: ISBN:

2 2 Preliminaries In all that follows Q m, c (z) will stands for the complexvalued function z m +c, where c is the complex parameter. The collection of points that are bounded, i.e., there exists a positive real number M such that Q n m,c(z) < M for all n, where n is the number of iterations, is called the prisoner set, while the collection of points that escapes to infinity is called the escape set. With reference to a Julia set, a Mandelbrot set consists of all those values of c for which the filled Julia set is connected (Shishikura [29], see also [25]). In all that follows, X will denote the set of complex numbers. Let ƒ: X->X for a point x 0 X, constructs a sequence {x n } in X as follows: x n = β n ƒ(x n-1 ) + (1-β n )x n-1, n = 1, 2,..., where 0 β n 1 and {β n } is convergent to β away from 0. DEFINITION 2.1. The sequence {x n } constructed above and denoted by GSO (ƒ, x 0, βn) will be called the general superior orbit for ƒ:x->x with the initial choice x 0. The general superior orbit GSO (ƒ, x 0, βn) with βn=β, denoted by SO (ƒ, x 0, β), is called the superior orbit ([25] and [26]). We get the definition of the Picard orbit, when one takes β = 1 in the superior orbit. The sequence {x n } constructed above, due to W.R. Mann [16], was essentially introduced in nonlinear analysis. However, recently, Rani and Kumar [25] have used the same to study chaos and fractals in discrete dynamics. Following their approach, we shall study the Mandelbrot set on a more general setting. DEFINITION 2.2 (Rani and Kumar [26]). The superior Mandelbrot set (SM set) for the polynomial Q m, c (z) := z m + c, where m > 1 is a positive integer, is defined as the collection of values of c for which the superior orbit of z = 0 does not escape to infinity. Notice that the Mandelbrot sets for Q m, c (z) are SM sets with β = 1. The escape criterion plays a vital role in the generation and analysis of Mandelbrot sets and its variants. We shall need the following result, which gives a general escape criterion for the polynomial Q m, c (z), m 2. THEOREM. Let Q m, c (z ) = z m + c, where m = 2, 3,., and 0 < λ(n) 1. For some z 0 = z, define z 1 = (1 - λ(n))z + λ(n) Q m,c (z),., z n = (1 - λ(n)) z n-1 + λ(n) Q m, c (z n-1 ), n = 2, 3. Then the general escape criterion is max { c, (2/ λ(n)) 1/(m-1) }. Here λ(n) evidently means that λ is some function of n, and in particular λ(n) = β n. This result is essentially due to Rani and Kumar ([25] and [26]) when λ(n) = β. This theorem gives a general algorithm for computing SM sets for the functions of the form Q m, c (z) = z m + c, m = 2, 3,... 3 Generation of SM Sets To generate SM sets for quadratics, an algorithm may be found due to Rani ([25] and [26]). The same may be used for generating general SM sets. Here, we have generated SM sets for polynomials of the form Q m, c (z)= z m + c, where m 2. We have written our program in C++ for the same and presented some attractive figures. 3.1 SM set for Q 10, c (z) = z 10 + c We draw superior Mandelbrot sets for β = 0.4 and 0.7 in Fig. 2 and 3 respectively. For m = 2 and β = 1, this algorithm reduces to the Mandelbrot set (see Fig. 1). 3.2 SM set for Q 25,c (z) = z 25 + c See the SM sets for β = 0.5 and 0.9 in Fig. 4 and 5 respectively. 3.3 SM set for Q 100,c (z) = z c We have generated SM sets for z c, taking β = 0.6 and 1.0 in Fig. 6 and 7 respectively. Notice that the sequence of SM sets is gradually moving towards a shape akin to a circular saw. 3.4 SM set for Q 400,c (z) = z c In this section, we construct superior Mandelbrot sets when m = 400, for two different values β, viz., 0.35 and 0.9 in Figure 8 and 9 respectively. Notice that the shape of the figures look like a fine (irregular) circular saw General SM sets for Q 6,c (z) We also generated general SM sets or equivalently a sequence of SM sets by considering general values of {β n } in the general superior orbit for polynomials of the form Q m, c (z). In many generated SM sets, we consider only a few striking ones for a sequence of {β n } that converges to β. For example, taking β n = /log (n+3), we depict different figures for various values of n. Notice that 0 β n 1 and β n β = 0.4. In this discussion, due to limitation of space, we take n = 100, 500, 1000, 1000 (see, Fig. 10, 11, 12, 13). Notice that as n increases, the graphics shows gradual convergence of general SM sets to an SM set (see Fig. 14). 4 Conclusion In this experimental study we generated numerous SM sets and noticed some striking features. Here, we presented only a few significant figures. We observe that an SM set for a polynomial of degree m has (m - 1) prime bulbs. For example, SM sets for polynomials of degree 10 and 25 contain 9 and 24 prime bulbs ISSN: ISBN:

3 respectively (see Fig. 2 and 4). It suggests that as the degree m becomes larger, the number of prime bulbs increases accordingly and the SM sets appear to be a circular saw or teething ring, and bulbs looks like teeth (see Fig. 6 and 8). We also notice from graphics of the figures that teeth become sharper as β becomes larger in z m + c. Also we find graphically that a sequence of general superior Mandelbrot sets converges to a superior Mandelbrot set. Fig. 3: Superior Mandelbrot Set for Q 10,c (z) with β = 0.7 Fig. 1: Mandelbrot Set for Q 2,c (z) with β = 1 Fig. 4: Superior Mandelbrot Set for Q 25,c (z) with β = 0.5 Fig. 2: Superior Mandelbrot Set for Q 10,c (z) with β = 0.4 Fig. 5: Superior Mandelbrot Set for Q 25,c (z) with β = 0.9 ISSN: ISBN:

4 Fig. 6: Superior Mandelbrot Set for Q 100,c (z) with β = 0.6 Fig. 7: Superior Mandelbrot Set for Q 100,c (z) with β = 1.0 ISSN: ISBN:

5 Fig. 8: Superior Mandelbrot Set for Q 400,c (z) with β = 0.35 Fig. 11: General Superior Mandelbrot Set for Q 6,c (z) with β n = /log (n + 3) and n = 500 Fig. 9: Superior Mandelbrot Set for Q 400,c (z) with β = 0.9 Fig. 12. General Superior Mandelbrot Set for Q 6,c (z) with β n = /log (n + 3) and n = 1000 Fig. 10: General Superior Mandelbrot Set for Q 6,c (z) with β n = /log (n + 3) and n = 100 Fig. 13. General Superior Mandelbrot Set for Q 6,c (z) with β n = /log (n + 3) and n = ISSN: ISBN:

6 Fig. 14. Superior Mandelbrot Set for Q 6,c (z) with β n = 0.4 References: [1] John Argyris, Ioannis Andreadis, and Theodoros E. Karakasidis, On perturbation of the Mandelbrot map, Chaos, Solitons & Fractals (2000)(11)(7), pp [2] John Argyris, Theodoros E. Karakasidis and Ioannis Andreadis, On the Julia sets of a noise-perturbed Mandelbrot map, Chaos, Solitons, & Fractals (2000)(13)(2), pp [3] D. Ashlock, Evolutionary Exploration of the Mandelbrot Set, Proc. IEEE Congress on Evolutionary Computation, CEC 2006, pp [4] B. Branner and J. Hubbard, The iteration of cubic polynomials, Part I, Acta. Math. 66, 1988, [5] A. J. Crilly, R. A. Earnshaw and H. Jones, Fractals and Chaos, Springer-Verlag, New York, Inc., [6] Richard M. Crownover, Introduction to Fractals and Chaos, Jones & Barlett Publishers, [7] Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, The Benjamin/Cummings Publishing Co. Inc., [8] Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison- Wesley, [9] Jagannathan Gomatam, John Doyle, Bonnie Steves, and Isobel McFarlane, Generalization of the Mandelbrot set: Quaternionic quadratic maps, Chaos, Solitons & Fractals (1995)(5)(6), pp [10] C. J. Griffin and G. C. Joshi, Transition points in octonionic Julia sets, Chaos, Solitons & Fractals (1993)(3)(1), pp [11] Jun Kigami, Analysis on Fractals, Cambridge Univ. Press, Cambridge, [12] Andrew Kricker and Girish Joshi, Bifurcation phenomena of the non-associative octonionic quadratic, Chaos, Solitons & Fractals (1995)(5)(5), pp [13] S. Kumar, Public Key Cryptographic System Using Mandelbrot Sets, Proc. IEEE conference on Military Communications Conference, MILCOM 2006, pp [14] Tan Lei, Similarity between the Mandelbrot sets and Julia sets, Commun. Math. Phys. 134, 1990, pp [15] Benoit Mandelbrot, Self-affine Fractals and Fractal Dimension, Physica Scripta 32, 1985, pp [16] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4, 1953, pp [17] M. S. El Naschie, Maximum entropy, information without probability and complex fractals: classical and quantum approach; Guy Jumarie (Ed.); Kluwer Academic Publishers, Dordrecht, 2000, Chaos, Solitons & Fractals (2001)(12)(6),pp [18] Ashish Negi and Mamta Rani, A new approach to dynamic noise on superior Mandelbrot set, Chaos, Solitons & Fractals (2008)(36)(4), pp [19] Ashish Negi and Mamta Rani, Midgets of superior Mandelbrot set, Chaos, Solitons & Fractals, (2008)(36)(2), pp [20] G. Pastor, M. Romera, G. Álvarez, D. Arroyo, and F. Montoya, On periodic and chaotic regions in the Mandelbrot set, Chaos, Solitons & Fractals (2007)(32)(1), pp [21] G. Pastor, M. Romera, G. Alvarez, and F. Montoya, External arguments for the chaotic bands calculation in the Mandelbrot set, Physica A: Statistical Mechanics and its Applications (2005)(353)(1), pp [22] G. Pastor, M. Romera, G. Álvarez, and F. Montoya, Chaotic bands in Mandelbrot set, Computers & Graphics (2008)(28)(5), pp [23] H. O. Peitgen, Jurgens and Saupe, Chaos and Fractals, Springer-Verlag, New York, Inc., [24] H. O. Peitgen, D. Saupe (Eds), The Science of Fractal Images, Springer-Verlag, [25] Mamta.Rani, Iterative procedures in Fractals and Chaos, Ph. D. Thesis, Department of Computer Science, Faculty of Technology, Gurukala Kangri Vishvavidyala, Hardwar, [26] M. Rani and V. Kumar, Superior Mandelbrot set, J. Korean Soc. Math. Edu. Ser. D (2004)8(4), pp [27] D. Rochon, A generalized Mandelbrot set for bicomplex numbers, Fractals 8(4)(2000), pp [28] M. Romera, G. Pastor, and F. Montoya, On the cusp and the tip of a midget in the Mandelbrot set antenna, Physics Letters A (1996) (221) (3-4),pp [29] M. Shishikura, Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math 147 (1998), no. 2, MR [30] Xing-yuan Wang, Pei-jun Chang, and Ni-ni Gu, Additive perturbed generalized Mandelbrot Julia sets, Applied Mathematics and Computation (2007)(189)(1), pp ISSN: ISBN: