Sunil Shukla et al, Int.J.Computer Technology & Applications,Vol 4 (2), STUDY OF NEWBIE FRACTAL CONTROLLED BY LOG FUNCTION

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1 ISSN: STUDY OF NEWBIE FRACTAL CONTROLLED BY LOG FUNCTION Sunil Shukla Ashish Negi Department of Computer Science Department of Computer Science & Engineering Omkarananda Institute of Management G.B Pant Engineering College & Technolog, Rishikesh Pauri Garhwal, Tehri Garhwal, Abstract In this paper we have presented the comple dnamics of Newbie fractal controlled b log function using Superior iterates. We have introduced the new fractal based on new function named as newbie fractal. This function is the variant of Mandelbrot function. These fractals are governed b initial controlled functions such as sin, cos, log, tan, conj, abs etc. Ke words: Comple dnamics, Newbie Fractal 1. Introduction Fractal geometr is based on the idea of selfsimilar forms. In this paper we have presented the stud of variants of escape time fractals. For this purpose we have taken one special function described as follows: Modified piel coordinate of C-plain b augmenting it with a set of given function 1 i.e. c fn1 p1. For the analsis piel purpose we have presented the stud of comple dnamics of the new fractal defined above, with log function.. Generation process The fractals have been generated b iterative formula z n1 f ( zn) where z0 is initial value of z, and z i is the value of the comple quantit z. The Mandelbrot s selfsquared function for generating fractals is f z z c p z and c are both p ( ),, comple quantities. We use of the transformation p ( ), function f z z c p for generating fractal images with respect to superior iterates, where z and c are the comple quantities and the power p is real number. These fractal IJCTA Mar-Apr

2 ISSN: images are constructed as a two-dimensional arra of piels. Each piel is represented b a pair of (, ) co-ordinates. The comple quantities z and c can be represented as: z z iz and c c ic, where i ( 1) and z, c are the real parts and z & c are the imaginar parts of z and c respectivel. The piel coordinates (, ) ma be associated with ( c, c) or ( z, z ) which is based on this concept, the fractal images can be classified as follows: a) z- plane fractals, wherein (, ) is a function of ( z, z ). b) c -plane fractals, wherein ( is, ) a function of ( c, c). In the literature, the fractals for p in z plane are termed as the Mandelbrot set while the fractals for p in c plane are known as Julia sets [1-5]. 3. Julia set French mathematician Gaston Julia [5] investigated the iteration process of a comple function intensivel, and attained the Julia set, a ver important and useful concept. At present Julia set has been applied widel in computer graphics, biolog, engineering and other branches of mathematical sciences [] and [8]. Consider the comple-valued quadratic function z z c c C, where c be the set of n1 n ; comple numbers and n is the iteration number. The Julia set for parameter c is defined as the boundar between those of z0 that remain bounded after repeated iterations and those escape to infinit. The Julia set on the real ais are reflection smmetric, while those with comple parameter show rotation smmetr with an eception to c (0, 0) see Rani and Kumar [6] and [11]. 4. Mandelbrot set The Mandelbrot set M for the quadratic Q z z c () c is define as the collection of all c C for which the orbit of the point 0 is bounded, i.e n M { c C :{ Q (0)} is bounded }}. 5. Superior iterates c n0,1,,... Let A be a subset of real or comple numbers and f : A A. For 0 A, construct a sequence { n } in A in the following manner 1 1 s f s s f s s f s n n n1 n n1 IJCTA Mar-Apr

3 ISSN: where 0 s n 1 and s is convergent to a non-zero number.the sequence n constructed above is called Mann sequence of iterates or superior sequence of iterates. Let z 0 be an arbitraril element of C, construct a sequences z of points of C in the following manner: n n z sf z 1 s z n 1,,3..., n n1 n1, where f is a function on a subset of C and the parameter s lie in the closed interval [0, 1].The sequence z n denoted b,, 0 constructed above, SO f z s is superior orbit for the comple-valued function f with an initial choice z0 and parameter s. We ma denote it b SO f s, 0, n, 0, n SO f s with 1 n. Notice that s is O f,. 0 We For log controlled functions we obtain a pair of opposite faced mini Mandelbrot bulbs. On increasing the value of p 1 and keeping p to, we observe the number of mini Mandelbrot bulbs of order p 1 see Fig Whereas, on increase the value of p and keeping p 1 to, we obtain the mini Mandelbrot bulbs of order p -1, see Fig We observe a beautiful triangular, full of spiral superior Julia sets at p 1 =3, p = and s=0.1, see Fig We also obtain the superior Julia set with distortion at p 1 =3, p = and s=0. see Fig. 11. Further the triangular collage of dendrites in a triangles is formed at p 1 = 3, p = and s = 0.5, see Fig. 1. remark that the superior orbit reduces to the usual Picard orbit when sn Analsis of superior function controlled julia set NF Mandelbrot set and Julia set for log controlled function: IJCTA Mar-Apr

4 ISSN: Fig. 1. For D (z), fn 1 = log, p 1 =, p = Fig. 4. For D (z), fn 1 = log, p 1 =, p = 3 Fig. 5. For D (z), fn 1 = log, p 1 =, p = 4 and s=1 Fig.. For D (z), fn 1 = log, p 1 = 3, p = Fig. 6. For D (z), fn 1 = log, p 1 =, p = 6 and s = 1 Fig. 3. For D (z), fn 1 = log, p 1 = 5, p = Fig. 7. For D (z), fn 1 = log, p 1 =, p = 6 IJCTA Mar-Apr

5 ISSN: (Zoom of Fig 51. Note the number of ovoid in outer Mandelbrot) Fig. 10. For D (z) = (0.865, -0.0), fn 1 = log, p 1 =3, p = and s = 0.1 Fig. 8. For D (z), fn 1 = log, p 1 = 3, p = and s = 0.1 Fig. 11. For D (z) = (.74, -1.33), fn 1 = log, p 1 =3, p = and s = 0. Fig. 9. For D (z) = (1.75, ), fn 1 = log, p 1 = 3, p = and s = 0.1 Fig. 1. For D (z) = (-0.85,.95), fn 1 = log, p 1 =3, p = and s= CONCLUSION In this paper we have presented the analsis of superior function controlled julia set using log function. Further we have analsis superior iterates at different power of p 1 and p see Fig Further we have presented the geometric properties of superior Julia sets along different ais. We IJCTA Mar-Apr

6 ISSN: also found two interesting image which resemble to beautiful triangular, full of spiral and collage of dendrites in a triangle see Fig. 10 and 1. REFERENCES 1. Barcellos, A. and Barnsle, Michael F., Reviews: Fractals Everwhere. Amer. Math. Monthl, No. 3, pp , Barnsle, Michael F., Fractals Everwhere. Academic Press, INC, New York, Edgar, Gerald A., Classics on Fractals. Westview Press, Falconer, K., Techniques in fractal geometr. John Wile & Sons, England, Julia, G., Sur 1 iteration des functions rationnelles. J Math Pure Appl. pp Kumar, Manish. and Rani, Mamta., A new approach to superior Julia sets. J. nature. Phs. Sci, pp , Negi, A., Fractal Generation and Applications, Ph.D Thesis, Department of Mathematics, Gurukula Kangri Vishwavidalaa, Hardwar, Orsucci, Franco F. and Sala, N., Chaos and Compleit Research Compendium. Nova Science Publishers, Inc., New York, Peitgen, H. O., Jurgens, H. and Saupe, D., Chaos and Fractals. New frontiers of science, Peitgen, H.O., Jurgens, H. and Saupe, D., Chaos and Fractals: New Frontiers of Science. Springer-Verlag, New York, Inc, Rani, M., Iterative Procedures in Fractal and Chaos. Ph.D Thesis, Department of Computer Science. Gurukula Kangri Vishwavidalaa, Hardwar, 00 IJCTA Mar-Apr