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International Journal of JOURNAL OF and Technology (IJCET), ISSN 0976- INTERNATIONALComputer EngineeringCOMPUTER ENGINEERING 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME & TECHNOLOGY (IJCET) ISSN 0976 – 6367(Print) ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), pp. 263-274 IJCET © IAEME: www.iaeme.com/ijcet.asp Journal Impact Factor (2012): 3.9580 (Calculated by GISI) ©IAEME www.jifactor.com COMPLEX DYNAMICS OF SUPERIOR PHOENIX SET Sunil Shukla *, Ashish Negi ** * Department of Computer Science Omkarananda Institute of Management & Technology, Rishikesh Tehri Garhwal, 249192. Email: shuklasunil@rediffmail.com ** Department of Computer Science & Engineering G.B Pant Engineering College, Pauri Garhwal, 246001. Email: ashish_ne@yahoo.com ABSTRACT The Phoenix fractal is a variant of the classic Mandelbrot and Julia sets. The Phoenix (Julia) type is particularly interesting, with beautiful shapes and lots of spirals. The Phoenix function, first introduced by Shigehiro Ushiki, is given by complex function zn +1 = zn p + real (c) + img (c) zn −1 , p ≥ 2 and n & c are constants. The study of Ushiki shows that the phoenix set does not have the same Mandelbrot and Julia Set properties as the classic Mandelbrot Set. In this paper we have presented different characteristics of phoenix function using superior iterates. Further, different properties like trajectories, fixed point, its complex dynamics and its behaviour towards Julia set are also discussed in the paper. Key words: Complex dynamics, Phoenix 1. INTRODUCTION Julia sets [1] and [9-10] provide a most striking illustration of how an apparently simple process can lead to highly intricate sets. Function on the complex plain c as simple as z n = z n 2 + c give rise of fractals of an exotic appearance [1]. This function zn for complex c has many fascinating mathematical properties and produces a wide range of interesting images [2], [3-5] and [9-10]. The superior iterates introduced by Rani and Kumar [6] and [11] in the study of chaos and fractal were found to be very effective in generating the fractals beyond the traditional limits. The Phoenix function was introduced by Shigehiro Ushiki [12] using complex function z n + 1 = z n 2 + r e a l ( c ) + im g ( c ) z n − 1 , where n and c are constants. 263 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME The complex dynamics of Phoenix function, generally known as Phoenix fractal, is a modification of the classic Mandelbrot and Julia sets. The study of Ushiki shows that the phoenix set does not have the same Mandelbrot and Julia Set properties as the classic Mandelbrot Set [1] and [3]. The Phoenix (Julia) type is particularly interesting, with beautiful shapes and lots of spirals. In this paper we have presented different characteristics of phoenix function using superior iterates and generated the superior Phoenix fractal. Further, different properties like trajectories, fixed point, its complex dynamics and its behaviour towards Julia set are also discussed in the paper. 2. PRELIMINARIES Definition 2.1. Julia sets French mathematician Gaston Julia [2] and [4-5] investigated the iteration process [2] of a complex function intensively, and attained the Julia set, a very important and useful concept. At present Julia sets has been applied widely in computer graphics, biology, Engineering and other branches of mathematical sciences. Consider the complex-valued quadratic function zn +1 = zn 2 + c, c ∈ C , where C be the set of complex numbers and n is the iteration number. The Julia set for parameter c is defined as the boundary between those of z0 that remain bounded after repeated iterations and those escape to infinity. The Julia set on the real axis are reflection symmetric, while those with complex parameter show rotation symmetry with an exception to c = (0, 0), see Rani and Kumar [6], [7] and [11]. Definition 2.2. Superior Orbit Let A be a subset of real or complex numbers and f : A → A. For x0 ∈ A, construct a sequence { xn } in A in the following manner x1 = s1 f ( x0 ) + (1 − s1 ) x0 x2 = s2 f ( x1 ) + (1 − s2 ) x1 M xn = sn f ( xn −1 ) + (1 − sn ) xn −1 Where 0 < sn ≤ 1 and {s n } is convergent to a non-zero number. The sequence { x n } constructed above is called Mann sequence of iterates or superior sequence of iterates. Let z0 be an arbitrarily element of C , Construct a sequences { z n } of points of C in the following manner: zn = sf ( z n −1 ) + (1 − s ) zn −1, n = 1, 2,3...., 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 } constructed above, denoted by SO ( f , z 0 , s ) is superior orbit for the complex-valued function f with an initial choice z0 and parameter s . We may denote it 264 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME by SO ( f , x0 , sn ) . Notice that SO ( f , x0 , sn ) with sn = 1 is O ( f , x0 ) . We remark that the superior orbit reduces to the usual Picard orbit when sn = 1. Definition 2.3. Henon map The Henon map is a prototypical 2-D invertible iterated map with chaotic solutions proposed by the Michel Henon [8-9], see Fig. 1. xn +1 = 1 + axn 2 + byn yn +1 = xn The values used to produce chaotic solutions are a = -1.4, b = 0.3. Fig. 1. Henon Map in real Plane Definition 2.4. Phoenix set The Phoenix function was introduced by the Shigehiro Ushiki [12], using complex function z n + 1 = z n p + r e a l ( c ) + im g ( c ) z n − 1 , where p ≥ 2 and n & c are constants. The complex Fig. 2. Phoenix Fractal in complex plane dynamics of Phoenix function, generally known as Phoenix fractal, is a modification of the classic Mandelbrot and Julia sets. The study of Ushiki shows that the phoenix set does not have the same properties as the classical Mandelbrot Set see Fig. 2, represents the complex one-dimensional section of a “Julia-like” set of a complexified “Henon map”. Define a 2 2 holomorphic automorphism of the two dimensional complex Euclidean space f : c → c by 265 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME f ( x, y) = ( x2 + c + by, x) , where c and b are the complex constants, see [12]. The phoenix appears when the attractive fixed point of this mapping loses its stability via a saddle-node bifurcation. The parameter values are chosen as b = -0.5, c = 0.56667. The picture represents 2 a complex line {x = y} in c , with n ranging from -0.5 ≤ Re(x) ≤ 1.2 (vertical) and -1.2 ≤ Im(x) ≤ 1.2 (horizontal). Definition 2.5. Superior Phoenix set The sequence { xn } constructed above is called Mann sequence of iteration or superior sequence of iterates. We may denote it by SO ( f , xo , sn ) . Now we define the Mandelbrot set for zn +1 = zn p + real (c) + img (c) zn−1 , where p ≥ 2 and n = 2, 3, 4,... with respect to Mann iterates. The collection of points whose orbits are bounded under the superior iteration for the Phoenix function, described above, is called the filled superior Phoenix set. 3. ANALYSIS In this section we have presented the Complex dynamics of Julia sets of Phoenix function using superior iterates. Further, we have also presented the convergence of phoenix function for different values of s and c. For z0 = (-0.124, 1.61) and s = 0.3, we observe that the value for F (z) converge to a fixed point i.e. 0.77086, see Table 1 and Fig. 3. On increasing the value to z0 = (-0.124, 1.61) and s = 0.5 we obtain two fixed points i.e. 0.4402 and 0.9908 see Table 2 & Fig. 4. Further, on increasing the value to z0 = (-5.8347, 0.1359) and s = 0.4 we find two fixed points i.e. 2.5896 and 1.2745 see Table 4 & Fig. 6. On increasing the value of z0 to (-55, 0) and fixing at s to 0.14, we obtain two fixed points i.e. 4.5863 and 8.6995, see Table 6 & Fig. 8. For z0 = (1.4246,-1.3085) and s = 0.3 we observe that the function escape to infinity, see Table 3 & Fig. 5. For z0 = (0.4868, 1.1694) and s = 0.5 we find that the value is escape to infinity, see Table 5 & Fig. 7. Number of iteration i |F(z)| 1 0.001 2 0.00 19 0.62668 20 0.64864 54 0.77067 55 0.7707 56 0.77073 57 0.77075 73 0.77085 74 0.77086 75 0.77086 76 0.77086 Table 1. F (z) for (z0 = -0.124, 1.61) at s = 0.3 (Some intermediate iteration has been skipped intentionally) 266 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME Fig. 3. F (z) for (z0 = -0.124, 1.61) at s = 0.3 (We skipped 73 iterations and after 74 iterations value converges to a fixed point.) Number of iteration i | F(z)| 1 0.001 2 0 42 0.76816 43 0.77344 125 0.846 126 0.68146 210 0.44008 211 0.94974 250 0.4402 251 0.9498 252 0.4402 253 0.9498 Table 2. F (z) for (z0 = -0.124, 1.61) at s = 0.5 (Some intermediate iteration has been skipped intentionally) Fig. 4. F (z) for (z0 = -0.124, 1.61) at s = 0.5 (We skipped 250 iterations and after 251 iterations value converges to two fixed point) 267 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME Number of iteration i |F(z)| 1 0.001 2 0 3 0.42738 4 0.78134 5 0.9897 16 2.0454 17 2.3984 28 2.33E+111 29 1.63E+222 30 NaN 31 NaN 32 NaN Table 3. F (z) for (z0 = 1.4246,-1.3085) at s = 0.3 (Some intermediate iteration has been skipped intentionally) Fig. 5. F (z) for (z0 = 1.4246,-1.3085) at s = 0.3 (We skipped 29 iterations and after 30 iterations value converges to infinity) Number of iteration i |F(z)| 1 0.001 2 0 3 2.3339 10 1.4435 11 2.5013 41 2.5887 42 1.276 43 2.5889 61 2.5896 62 1.2745 63 2.5896 64 1.2745 Table 4. F (z) for (z0 =-5.8347, 0.1359) at s = 0.4 (Some intermediate iteration has been skipped intentionally) 268 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME Fig. 6. F (z) for (z0 = -5.8347, 0.1359) at s = 0.4 (We skipped 61 iterations and after 62 iterations value converges to two fixed point) Number of iteration i |F(z)| 1 0.001 2 0 3 0.2434 4 0.39472 5 0.66098 6 1.0231 7 1.6648 16 1.09E+79 17 5.90E+157 18 NaN 19 NaN 20 NaN Table 5. F (z) for (z0 = 0.4868, 1.1694) at s = 0.5 (Some intermediate iteration has been skipped intentionally) Fig. 7. F (z) for (z0 = 0.4868, 1.1694) at s = 0.5 (We skipped 17 iterations and after 18 iterations value converges to infinity) 269 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME Number of iteration i |F(z)| 1 0.001 2 0 3 7.7 10 5.5894 11 8.1331 57 8.6993 58 4.5865 59 8.6994 74 4.5863 75 8.6995 76 4.5863 77 8.6995 Table 6. F (z) for( z0 = -55, 0) at s = 0.14 (Some intermediate iteration has been skipped intentionally) Fig. 8. F (z) for z0 = (-55, 0) at s = 0.14 (We skipped 17 iterations and after 18 iterations value converges to infinity) 4. GENERATION OF SUPERIOR JULIA SETS FOR PHOENIX SET Here we have presented some beautiful filled relative superior Julia sets for the phoenix function. In most of the figures we found symmetry along x axis. As an exception we found some Phoenix Julia sets symmetrical around x as well as y axis see Fig. 12. It is observed that the orbit of Phoenix function converges to either 1 or 2 point. It is observed that the superior Julia sets for Phoenix function to be symmetric along x axis for even powers, see Fig. 15-16, and symmetric along y axis for the odd powers of p see Fig. 17-18. 270 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME Fig. 9 Superior Phoenix set for (z0 = -0.124, 1.61) at s = 0.3, p = 2 Fig. 10. Superior Phoenix set for (z0 = -0.124, 1.61) at s = 0.5, p = 2 Fig. 11. Superior Phoenix set for (z0 = 1.4246,-1.3085) at s = 0.3, p = 2 Fig. 12. Superior Phoenix set for (z0 = -5.8347, 0.1359) at s = 0.4, p = 2 271 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME Fig. 13. Superior Phoenix set for (z0 = 0.4868, 1.1694) at s = 0.5, p = 2 Fig. 14. Superior Phoenix set for (z0 = -7, -9) at s = 0.1, p = 2 Fig. 15. Superior Phoenix set for (z0 = 0.9812, 1.9233) at s = 0.1, p = 4 Fig. 16. Superior Phoenix set for (z0 = -0.55, 0.931) at s = 0.3, p = 12 272 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976- 6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME Fig. 17. Superior Phoenix set for (z0 = 0.006, -1.118) at s = 0.9, p = 3 Fig. 18. Superior Phoenix set for (z0 = 0.006, -1.118) at s = 0.9, p = 3 5. CONCLUSION In this paper we have presented the dynamics and fixed point analysis of Phoenix set by using superior Iterates. Further we have presented the geometric properties of superior Julia sets for Phoenix function along different axis. We have also presented an image which resemble to a pair of leaf, see Fig. 11 and famous spider fractal see Fig. 10. 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