Generation of Anti-Fractals in SP-Orbit

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1 International Journal of Coputer Trends and Tehnology (IJCTT) Volue 43 Nuber 2 January 2017 Generation of Anti-Fratals in SP-Orbit Mandeep Kuari 1, Sudesh Kuari 2, Renu Chugh 3 1,2,3 Departent of Matheatis, Maharshi Dayanand University, Rohtak-12400, Haryana, India Abstrat In this paper we generate a new lass of Triorns and Multiorns using SP iteration (a fourstep feedbak proess) and explore the geoetry of superior antifratals. Other researhers have already generated antifratals using Piard, Mann, ishikawa and Noor orbits that are exaples of one step, two-step, three-step and four-step feedbak proesses. Keywords Antipolynoial, antifratal, Triorn, Multiorn, SP-orbit MSC: 37F45, 37F50. I. INTRODUCTION Fratals are defined as objets that appear to be broken into a nuber of piees and eah piee is a opy of the entire shape. Fratal is the word taken fro the Latin word fratus whih eans broken. The ter fratal was first used by a young atheatiian, Mandelbrot [2]. Julia introdued the onept of iterative funtion and he derived the Julia set in After that, in 1979, Mandelbrot [2] extended the work of Gaston Julia and introdued the Mandelbrot set, a set of all onneted Julia sets. Many researhers have studied Julia sets and Mandelbrot sets fro different aspets. The onneted lous of antipolynoial z z is known as Triorn. The ter Triorn was firstly used by Milnor. In 2003, Shizuo et al.[15] desribed various properties of Triorn and Multiorn by oputing beautiful figures and quoted that Multiorns are the generalized Triorns or the Triorns of higher order. The dynais of antiholoorphi oplex polynoials z z, for 2, was studied and explored to visualize interesting Triorns and Multiorns antifratals with respet to one-step feedbak proess [12], two step-feedbak proess [10, 11], three-step feedbak proess [18] and four step feedbak proess[1,3]. The dynais of antipolynoial z z where 2 with respet to iterative funtion generates aazing Triorn and Multiorns [12, 14, 15]. Crowe et. al. [16] onsidered it as a foral analogy with Mandelbrot sets and naed it as Mandelbar set. They also brought their bifuration features along ars rather than at points. Multiorns have been found in a real slie of the ubi onnetedness lous [15]. Winter [13] showed that the boundary of the Triorn ontains ar. The syetries of Triorn and Multiorns have been analyzed by Lau and Shleiher [4]. In 2011, W. Phuengrattana and S. Suantai [18] proposed the SP-iteration for approxiating a fixed point of ontinuous funtions on an arbitrary interval. They opared the onvergene speed of Mann, Ishikawa, Noor and SP-iterations using soe nuerial exaples and proved that the SP-iteration is equivalent to and onverges faster than the other iterations. In this paper we generate a new lass of Triorns and Multiorns under SP orbit whih is an exaple of four-step feedbak proess and analyze the. II. PRELIMINARIES Definition 1. [12] (Multiorn). The ultiorns A z z is defined as the olletion of all C for whih the orbit of the point 0 is bounded, that is for the quadrati funtion A : n C A 0 does not tend to where C is a oplex spae. z A n A is the n th iterate of the funtion A. An equivalent forulation is that the onnetedness of loi for higher degree antiholoorphi polynoials A z z are alled ultiorns. Note that at = 2, ultiorns redue to triorn. Naturally, the triorns lives in the real slie d in the two diensional paraeter spae of aps 2 2 z z d. They have (+1)-fold rotational syetries. Also, by dividing these syetries, the resulting ultiorns are alled uniorns [14]. Definition 2. [6] (Julia Set). The filled in Julia set of the funtion g is defined as K(g) = {z C: g k (z) does not tend to }, where C is the oplex spae, g k (z) is k th iterate of funtion g and K(g) denotes the filled Julia set. The Julia set of the funtion g is defined to be the boundary of K(g), i.e., J(g) = K(g), where J(g) denotes the Julia set. Definition 3. [12] (Mandelbrot Set). The Mandelbrot set M onsists of all paraeters for whih the filled Julia set of 2 Q z z is onneted, that is M = { C: K(Q ) is onneted }. ISSN: Page 105

2 International Journal of Coputer Trends and Tehnology (IJCTT) Volue X Issue Y- Month 2017 In fat, M ontains an enorous aount of inforation about the struture of Julia sets. The Mandelbrot set M for the Quadrati Q (z) = z 2 + is defined as the olletion of all C for whih the orbit of the point 0 is bounded, that is n M = { C: { Q (0)} ; n = 0, 1, 2, is bounded}. We hoose the initial point 0 as 0 is the only ritial point of Q. Now, we give definition of the SP orbit, whih will be used in the paper to ipleent four-step feedbak proess in the dynais of polynoial z z. Definition 4. [8] Let T : X X be a apping. Let us onsider a sequene {z n } of iterates for initial point z 0 X suh that {z n+1 : z n+1 = (1 α n ) u n + α n Tu n ; u n = (1 β n ) v n + β n Tv n ; v n = (1 n ) z n + n Tz n ; n = 0, 1, 2,...}, where α n, β n, n [0, 1] and {α n }, {β n }, { n } are sequenes of positive nubers. The above sequene of iterates is alled as SP orbit, whih is a funtion of five tuples (T, z 0, α n, β n, γ n ). III. MAIN RESULT Now, we will obtain a general esape riterion for polynoials of the for G ( z) z Theore 1. For a general funtion G () z z, =1, 2, 3, where 0 1, 0 1, 0 1, and is a oplex nuber. Define z1 (1 ) u G ( u) z2 (1 ) u1 G ( u1 ) z (1 ) u1 G ( u1), where = 1, 2, 3, 4, Then, the general Superior esape riterion 1/ 1 1/ 1 1/ 1 is ax{,(2 / ),(2 / ),(2 / ) }. Proof. For proving the theore, we shall use the ethod of indution. For = 1, we have G () z z, and this iplies z ax{, 0, 0, 0}. 2 For = 2, we have G () z z, then the esape riterion is z ax{,2 /,2 /,2 / }. 3 Siilarly, for = 3, we get G () z z. The esape riterion is z 1/2 1/2 1/2 ax{,(2 / ),(2 / ),(2 / ) }. Hene the theore is true for =1, 2, 3 Now, suppose that theore is true for any. We prove that the result is true for Let G () z z and z (2 / ) 1/, z (2 / ) 1/ and z (2 / ) 1/. Then, onsider v (1 ) z G ( z), 1 where G () z z (1 ) z ( z 1 ) z z 1 z z 1 z z i.e. v z z 1 Also, u (1 ) v G ( v) (1 ) v v 1 ( z ) (1) 1 (1 ) z ( z 1) z z 1...(2) Sine 2 1 z iplies z 1 1, (3) so z z 1 z. Using (3) in (2), we have 1 u 1 z z 1 z 1 z 1 z 1 z z ( z ) z z 1 i.e. u z z 1 (4) Now for z (1 ) u1 G ( u1), we have z1 (1 ) u G ( u) (1 ) u u 1 1 (1 ) z ( z 1) z z 1 ISSN: Page 106

3 International Journal of Coputer Trends and Tehnology (IJCTT) Volue X Issue Y- Month z 2 iplies z 1 1, Sine so z z 1 z. Using (6) in (5), we get 1 z1 1 z z 1, 1 z 1 z z (5) (6) We have the beautiful Rangoli Patterns (Figs. 16, 17). We also find that higher degree ultiorns beoe irular saw (Fig. 18). Soe authors [1,3,11] had also found the siilar onlusion while generating Multiorns using twostep, three-step, four-step feedbak proesses. The nae irular saw was, first, given by Rani and Kuar to Mandelbrot sets [9]. z z 1 i.e., z1 z z 1 Sine z (2 / ) 1/, z (2 / ) 1/ and z (2 / ) 1/ exist, we have z In partiular, z1 1 z 1 z Hene, z as. This opletes the proof. 1/ 1 Corollary 1.1. Suppose (2 / ), 1/ 1 (2 / ) and z 1/ 1 (2 / ) exists. Then the orbit SP( G,0,,, ) esapes to infinity. Corollary 1.2. (Esape Criterion). Let us Assue that for soe k 0, 1/ k 1 1/ k 1 1/ k1 z ax{,(2 / ),(2 / ),(2 / ) }, then z k k and z as. zk 1 This orollary gives an algorith to generate antijulia sets for the funtions of the type G () z z, = 2, 3, Thus, for visualizing new antifratals, the required esape riterion with respet to the SP orbit for z z is ax, 2 / 1, 2 / 1, 2 / 1 [7]. Fig.1: 0.3, 0.5, 0. 6 Fig.2: 0.6, 0.3, 0. 5 Fig. 3: 0.68, 0.27, 0.95 IV. MULTICORNS IN SP ORBIT All In this setion, we generate Triorns and Multiorns by prograing the polynoial z z in the software Matheatia 9.0 under SP orbit (see Figs. 1-18). We have the following observations: The nuber of branhes in the triorns and ultiorns is +1, where is the power of z. Also, few branhes have sub-branhes. The shapes of Triorns and Multiorns beoe different as we hange the values of paraeters. Fig. 4: 0.95, 0.27, 0.68 ISSN: Page 107

4 International Journal of Coputer Trends and Tehnology (IJCTT) Volue X Issue Y- Month 2017 Fig. 5: 0.9, 0.1, 0.1 Fig. 9: 3, 0.08, 0. 6 Fig. 6: 0.1, 0.1, 0.9 Fig.10: 3, 0.6, 0.08 Fig. 7: 0.5, 0.5, 0.5 Fig. 11: 3, 0.6, 0.08, 0.6 Fig.8 : 0.09, 0.7, 0.8 Fig.1 2 : 3, 0.5, 0.5, 0.5 ISSN: Page 108

5 International Journal of Coputer Trends and Tehnology (IJCTT) Volue X Issue Y- Month 2017 Fig. 13: 3, 0.3, 0.05, 0.6 Fig.17: 30, 0.05, 0.2 Fig.14: 4, 0.05, 0.05, 0.05 Fig.18: Cirular saw ultiorn for 75, 0.05 V. NEW ANTI JULIA SETS We opute anti Julia sets for z z in the software Matheatia 9.0 via SP orbit. We have the following observations while generating the. Fig. 15: 5, 0.05, 0.6, 0.3 In Figs.19-20, we notie that as we inrease the value of paraeters, and keeping onstant sae anti Julia sets beoe fattier. The nuber of branhes in anti Julia sets is +1, where is the power of z. Also, few branhes have sub-branhes (see Figs. 27, 28, 29). Also, we observe that the higher degree anti Julia sets take different shapes (like irular shaw and Rangoli pattern) for different values of,,, and. (see Figs ) Fig.16 : 15, 0.6, 0.6, Fig.19 : AntiJulia set for 2 0.3, I ISSN: Page 109

6 International Journal of Coputer Trends and Tehnology (IJCTT) Volue X Issue Y- Month 2017 Fig. 20 : AntiJulia set for 2 0.6, I Fig. 24 : AntiJulia set for 2 0.5, 0.9, 0.1, I Fig. 21: AntiJulia set for 2 0.3, 0.6, 0.9, I Fig. 25: AntiJulia set for 2 0.9, 0.1, 0.5, I Fig. 22 : AntiJulia set for 2 0.6, 0.3, 0.9, I Fig. 26 : AntiJulia set for 3 0.9, 0.1, 0.5, 0.1 Fig. 23: AntiJulia set for 2 0.9, 0.6, 0.3, I Fig. 27 : AntiJulia set for , I ISSN: Page 110

7 International Journal of Coputer Trends and Tehnology (IJCTT) Volue X Issue Y- Month 2017 Fig. 28: AntiJulia set for 3 0.1, 0.5, 0.9, 0.1 Fig. 32 : Cirular saw AntiJulia set for 50, 0.1, 0.5, 0.9, I Fig. 29 : AntiJulia set for 4 0.1, 0.5, 0.9, 0.1 VI. CONCLUSIONS In the dynais of antipolynoials z z, where > 2, there exist any antifratals for the sae value of but different values of paraeters in SP orbit. In our results, we find that for higher degree polynoials, all the antifratals beoe irular saw. We observe that Multiorns are syetrial about both x and y axis for odd values of, but for even values of, the syetry is aintained only along x-axis. ACKNOWLEDGMENT This work is supported by the University Grants Coission of India (Grant No. F.1763/2008(SA-1)). Fig. 30 : AntiJulia set for , 0.1, 0.5, 0.1 Fig. 31: AntiJulia set for , 0.9, 0.1, 0.1 REFERENCES [1] Ashish, M. Rani, and R. Chugh, Dynais of antifratals in Noor Orbit, International Journal of Coputer Appliations, Volue 57, Nuber 4 (2012) pp [2] B. B. Mandelbrot, The Fratal Geoetry of Nature, W. H. Freean, New York, NY, USA, [3] D. Negi and A. Negi, A behavior of Triorns and Multiorns in N-Orbit, International Journal of Applied Engineering Researh, Volue 11, Nuber 1 (2016) pp [4] E. Lau, and D. Shleiher, Syetries of fratals revisited, Math. Intelligener (18)(1)(1996), [5] G. Julia, Sur l iteration des funtions rationnelles, Journal de Math eatiques Pures et Appliqu ees, vol. 8, pp , [6] H. O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fratals, Springer-Verlag, New York, [7] M. Abbas and T. Nazir, A new iteration proess applied to onstained iniization and feasibility probles, Mateathyk Behnk (66)(2) (2014), [8] M. Kuari, Ashish and R. Chugh, New Julia And Mandelbrot Sets for a New Faster Iterative Proess, International Journal of Pure and Applied Matheatis, Volue-107, No. 1, 2016, [9] M. Rani, and M. Kuar, Cirular saw Mandelbrot sets, in: Pro. 14th WSEAS Int. Conf. on Appl. Math.(Math 09), 2009, [10] M. Rani, Superior antifratals, in: IEEE Pro. ICCAE 2010, vol. 1, ISSN: Page 111

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