Uttarkhand Technical University, J.B.Institute of Technology, Uttarakhand Technical University, Dehradun, INDIA Dehradun, INDIA Dehradun, INDIA

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Volume 3, Issue 12, December 2013 ISSN: 2277 128X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.

Tech. CSE Computer Science and Engg. Dept., Computer Science and Engg. Dept., Computer Science and Engg. Dept., Uttarkhand Technical University, J.B.

1 Volume 3, Issue 12, December 2013 ISSN: X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: Analysis of Fangled Mandelbrot and Julia Sets Controlled by Logarithmic Function Suraj Singh Panwar Kuldeep Singh Pawan Kumar Mishra M.Tech. - CSE, Scholar M.Tech.- CSE, Scholar Assistant Professor, M.Tech. CSE Computer Science and Engg. Dept., Computer Science and Engg. Dept., Computer Science and Engg. Dept., Uttarkhand Technical University, J.B.Institute of Technology, Uttarakhand Technical University, Dehradun, INDIA Dehradun, INDIA Dehradun, INDIA Abstract In this paper we explore the dynamics of complex logarithmic function for integer and non-integer values. We have used Ishikawa iteration method for generating fractals. In this paper we study and analyse the fractals generated using logarithmic function for Mandelbrot and Julia sets. The purpose of this paper is to generate the fractals using Z Z n +C, n>=2.0, C=log(1/#pixel p ), p>=1.0 for integer and non-integer values. The parameter s and s changes the structure and beauty of fractals. Different values of p and n also changes the characteristics of fractals for both integer and non-integer values. Keywords Fractals, Complex, Mandelbrot set, Julia set, Iteration technique, Ishikawa Iteration, Self-Similar, Computer Graphics. I. INTRODUCTION Fractal Theory is a fascinating branch of applicable Mathematics and Computer Science. It is based on the conception of self-similar forms. Benoit Mandelbrot ( ), scientist and mathematician is known as the father of fractal geometry. He coined the word fractal in late 1970s. He published to explain geometric fractals as a rough or fragmented geometric shape that can be divided into parts, every one of which is a reduced-size duplicate of the whole[1]. Fractals are all around us in the form of many usual objects such as mountains, coastlines, trees, ferns and clouds [7,8,13]. They all are fractals in nature and can be represented on a computer by a recursive algorithm of computer graphics. Before the innovation of computers, Fractals have appeared as an important question. Fractal is defined as a set, which is self similar under magnification [2]. In the perspective of complex dynamics, which is a topic of mathematics, the Julia sets and the Mandelbrot sets are defined from a function. The Julia sets and the Mandelbrot sets are two most important objects under various researches in the field of fractal theory [3]. The Julia set was given by French mathematician Gaston Julia ( ) in 1918, whereas the Mandelbrot set was given by Benoit B. Mandelbrot ( ) in There is a strong connection between the Mandelbrot sets and the Julia sets. For one Mandelbrot set there are many or we can say infinite Julia sets. II. PRELIMINARIES AND DEFINATIONS A. Mann s Iteration : One Step Iteration Iteration means to repeat a process again and again. Starting with the initial value, the output is fed back to the process. The procedure is repeated until the result or goal is reached. Mann s iteration technique is a one-step iteration technique given by William Robert Mann ( ), a mathematician from Chapel Hill, North Carolina. The iteration technique involves one step for iteration, and is given as [6,11]: x n+1 = (1-s)x n + s.f(x n ), where n 0 and 0 < s < 1 B. Ishikawa Iteration : Two Step Iteration Ishikawa iteration [11] technique is a two-step iteration method known after Ishikawa. Let X be a subset of complex number and f : X X for all x 0 ϵ X, we have the sequence numbers for {x n } and {y n } in X according to following way [4,5] : y S ' f ( x ) (1 S ' ) x n n n n n x 1 S f ( y ) (1 S ) x n n n n n Where, 0 S n 1, 0 S n 1 and S n & S n are both convergent to non-zero number. III. GENERATION OF RELATIVE SUPERIOR MANDELBROT SETS We present here some Relative Superior Mandelbrot sets [14,15] for the function Z Z n +C, n>=.02, C=log(1/#pixel p ), p>=1.0 for integer and some non-integer values. The parameter s and s also changes the structure and beauty of fractals. The value of p and n changes the characteristics of fractals for both integer and non-integer values. 2013, IJARCSSE All Rights Reserved Page 1198

A. Generation of Relative Superior Mandelbrot sets for

1 Relative Superior Mandelbrot set for s=s =1,p=2 for s=s =0.

3 Relative Superior Mandelbrot set for s=s =1,p=2.5 s=s =0.

5 Relative Superior Mandelbrot set for s=s =1,p=3 set for s=s

7 Relative Superior Mandelbrot set for s=s =1,p=3.8 Fig.

2 A. Generation of Relative Superior Mandelbrot sets for Quadratic function (i.e. Z 2 +C here n=2 and p varies). Fig. 1 Relative Superior Mandelbrot set for s=s =1,p=2 for s=s =0.5,p=2 Fig. 2 Relative Superior Mandelbrot set Fig. 3 Relative Superior Mandelbrot set for s=s =1,p=2.5 s=s =0.5,p=2.5 Fig. 4 Relative Superior Mandelbrot set for Fig. 5 Relative Superior Mandelbrot set for s=s =1,p=3 set for s=s =0.5,p=3 Fig. 6 Relative Superior Mandelbrot Fig. 7 Relative Superior Mandelbrot set for s=s =1,p=3.8 Fig. 8 Relative Superior Mandelbrot set for s=s =0.5,p= , IJARCSSE All Rights Reserved Page 1199

3 Fig. 9 Relative Superior Mandelbrot set for s=s =1,p=4.5 for s=s =.5,p=4.5 Fig. 10 Relative Superior Mandelbrot set Fig. 11 Relative Superior Mandelbrot set for s=s =1,p=8.5 Mandelbrot set for s=s =.5,p=8.5 Fig.12 Relative Superior B. Generation of Relative Superior Mandelbrot set for Cubic function ( i.e. Z 3 +C here n=3 and p varies). Fig. 13 Relative Superior Mandelbrot set for s=s =1,p=2 Mandelbrot set for s=s =.5,p=2 Fig. 14 Relative Superior Fig. 15 Relative Superior Mandelbrot set for s=s =1,p=2.5 for s=s =.5,p=2.5 Fig. 16 Relative Superior Mandelbrot set 2013, IJARCSSE All Rights Reserved Page 1200

4 Fig. 17 Relative Superior Mandelbrot set for s=s =1,p=3 Fig. 18 Relative Superior Mandelbrot set for s=s =.5,p=3 Fig. 19 Relative Superior Mandelbrot set for s=s =1,p=8 Fig. 20 Relative Superior Mandelbrot set for s=s =.5,p=8 C. Generation of Relative Superior Mandelbrot set for Bi-Quadratic function (i.e. Z 4 +C here n=4 and p varies). Fig. 21Relative Superior Mandelbrot set for s=s =1,p=2.5 Fig.22 Relative Superior Mandelbrot set for s=s =.5,p=2.5 Fig. 23 Relative Superior Mandelbrot set for s=s =1,p=3 Fig. 24 Relative Superior Mandelbrot set for s=s =.5,p=3 2013, IJARCSSE All Rights Reserved Page 1201

Fig. 25 Relative Superior Mandelbrot set

5 Fig. 25 Relative Superior Mandelbrot set for s=s =1,p=3.8 Fig. 26 Relative Superior Mandelbrot set for s=s =.5,p=3.8 Fig. 27 Relative Superior Mandelbrot set for s=s =1,p=8 Fig. 28 Relative Superior Mandelbrot set for s=s =.5,p=8 Fig. 29 Relative Superior Mandelbrot set for s=s =.5,p=11 Fig.30 Relative Superior Mandelbrot set for s=s =.5,p=11.5 D. Generalization of Relative Superior Mandelbrot sets (for different values of n and p).:- Fig.31 Relative Superior Mandelbrot set for s=s =1,p=4,n=4 Fig.32 Relative Superior Mandelbrot set for s=s =1,p=7,n=7 2013, IJARCSSE All Rights Reserved Page 1202

Fig.33 Relative Superior Mandelbrot set for s=s =.

GENERATION OF RELATIVE SUPERIOR JULIA SETS We present here some Relative

$The parameter s and s also changes the structure and beauty of fractals.$ $The value of p and n changes the characteristics of fractals for both integer$ Relative Superior Julia sets for different values of p and n for above function.

Relative Superior Julia sets for different values of p and n for above function.

36 Relative Superior Julia set for s=s =1,p=3,n=2, z 0 =(-0.265625,- 0.

6 Fig.33 Relative Superior Mandelbrot set for s=s =.5,p=6,n=7 Mandelbrot set for s=s =.5,p=7,n=6 Fig.34 Relative Superior IV. GENERATION OF RELATIVE SUPERIOR JULIA SETS We present here some Relative Superior Julia sets [14,15] for the function Z Z n +C, n>=2.0, C=log(1/#pixel p ), p>=1.0 for integer and some non-integer values. The parameter s and s also changes the structure and beauty of fractals. The value of p and n changes the characteristics of fractals for both integer and non-integer values. A. Relative Superior Julia sets for different values of p and n for above function. Fig. 35 Relative Superior Julia set for s=s =1,p=2,n=2, Fig. 36 Relative Superior Julia set for s=s =1,p=3,n=2, z 0 =( , ) z 0 =( , ) Fig. 37 Relative Superior Julia set for s=.1,s =0,p=3,n=2, Fig. 38 Relative Superior Julia set for s=0.1,s =0,p=4,n=2, z 0 =( , ) z 0 =(1.625, ) 2013, IJARCSSE All Rights Reserved Page 1203

$5,s =0,p=6,n=4, z 0 =(-1.5625, 0.00694) V. CONCLUSION In the complex dynamics polynomial function z (z n +c), where n>=2.0, control logarithmic function c=log(1/#pixel P ), p>=1.0. The fractals generated with power p and n are found as rotationally symmetric.$ $For higher values of n, the central body is bifurcated into (n-1) lobes from left side. The controlling function log for value c exhibits new characteristics for the generating fractals.$

7 Fig. 39 Relative Superior Julia set for s=s =1,p=4,n=3, Fig. 40 Relative Superior Julia set for s=0.5,s =0,p=5,n=2, z 0 =(0.5,0.6) z 0 =(-0.85, 2.95) Fig. 41 Relative Superior Julia set for s=0.5,s =0,p=6,n=4, z 0 =( , ) V. CONCLUSION In the complex dynamics polynomial function z (z n +c), where n>=2.0, control logarithmic function c=log(1/#pixel P ), p>=1.0. The fractals generated with power p and n are found as rotationally symmetric. We have analysis superior iterates at different power of n and p see above fig. There are ovoids or bulbs attached with main body. The number of major secondary lobe is (n-1). For higher values of n, the central body is bifurcated into (n-1) lobes from left side. The controlling function log for value c exhibits new characteristics for the generating fractals. Here we have presented the geometric properties of fractals along different axis. The fractals generated depends on the parameter p, i.e. p image is generated for above log controlling function. For p=2, two mirror image (opposite faced) of mini Mandelbrot bulbs are obtained at an angle of For p=3, three mirror image of mini Mandelbrot bulbs at an angle of /3=120 0 are separated from each other. So for integer value of p there are p mirror image at an angle of /p. For non-integer value of p new image is created from left side step by step till the upper integer value is met. For even value of p, fractals are symmetrical along x and y-axis, while for odd values fractals are symmetrical along x- axis. We have also generated some superior function controlled Julia set using log function, which resemble to beautiful triangular, tetrahedron, full of spiral and dendrites in a triangle or other shapes. Further by using different iterative methods and more different functions we can create fractals and analyse their properties. REFERENCES [1] B. B. Mandelbrot, The fractal geometry of nature. Macmillan. ISBN , [2] B. B. Mandelbrot, Fractals and Chaos, Berlin: Springer. pp. 38, ISBN "A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension", [3] G. Edgar, Classics on Fractals, Boulder, CO: Westview Press. ISBN , [4] Y.S. Chauhan,, R.Rana and Ashish Negi, New Julia Sets of Ishikawa Iterates, International Journal of Computer Applications (IJCA), Volume 7, No.13, October [5] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc.44, (1974), pp [6] S. Agarwal and Dr.A.Negi, Midgets of Transcendental Superior Mandelbar Set, International Journal of Computer Science Issues (IJCSI), Vol. 9, Issue 4, No.3, July [7] M. F. Barnsley, R. L. Devaney, B. B. Mandelbrot, H. O. Peitgen, D. Saupe, and R. F. Voss, The Science of Fractal Images, Springer Verlag , IJARCSSE All Rights Reserved Page 1204

8 [8] M. Batty, Fractals - Geometry Between Dimensions, New Scientist ( Holborn Publishing Group) 105 (1450): 31, [9] R.L.Daveney, An Introduction to Chaotic Dynamical Systems, Springer-Verlag, New York. Inc [10] A. Negi and, M. Rani, Midgets of Superior Mandelbrot Set, Chaos, Solitons and Fractals, July [11] R.Rana, Y.S.Chauhan and Ashish Negi, Ishikawa Iterates for Logarithmic function, International Journal of Computer Applications (IJCA), Volume 15, No.5, February [12] Ashish, Negi, Generation of Fractals and Applications, Thesis, Gurukul Kangri Vishwavidyalaya, [13] H.Peitgen and P.H.Richter, The Beauty of Fractals, Springer-Verlag, Berlin, [14] M. Rani, Iterative procedures in fractals and chaos, Ph.D. Thesis, Department of Computer Science, Faculty of Technology, GurukulKangri Vishvavidyalaya, Hardwar, [15] M.Rani, V.Kumar, Superior Mandelbrot Set, J.Koreans Soc. Math. Edu. Ser. D (2004) 8(4), pp , IJARCSSE All Rights Reserved Page 1205

New Mandelbrot and Julia Sets for Transcendental Function

New Mandelbrot and Julia Sets for Transcendental Function Suraj Singh Panwar #1, Mr.Pawan Kumar Mishra *2 # M.Tech. - CSE, Scholar, Faculty of Technology, Computer Science and Engineering Department, Uttarakhand