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Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
Complex dynamics of superior phoenix set
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Complex dynamics of superior phoenix set

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  • 1. INTERNATIONALComputer EngineeringCOMPUTER ENGINEERING International Journal of JOURNAL OF and Technology (IJCET), ISSN 0976- 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.aspJournal Impact Factor (2012): 3.9580 (Calculated by GISI) ©IAEMEwww.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
  • 2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEMEThe complex dynamics of Phoenix function, generally known as Phoenix fractal, is amodification of the classic Mandelbrot and Julia sets. The study of Ushiki shows that thephoenix set does not have the same Mandelbrot and Julia Set properties as the classicMandelbrot Set [1] and [3]. The Phoenix (Julia) type is particularly interesting, with beautifulshapes and lots of spirals. In this paper we have presented different characteristics of phoenixfunction using superior iterates and generated the superior Phoenix fractal. Further, differentproperties like trajectories, fixed point, its complex dynamics and its behaviour towards Juliaset are also discussed in the paper.2. PRELIMINARIESDefinition 2.1. Julia sets French mathematician Gaston Julia [2] and [4-5] investigated the iteration process [2] ofa complex function intensively, and attained the Julia set, a very important and usefulconcept. 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 forparameter c is defined as the boundary between those of z0 that remain bounded afterrepeated iterations and those escape to infinity. The Julia set on the real axis are reflectionsymmetric, while those with complex parameter show rotation symmetry with an exceptionto 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 asequence { 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 superiorsequence of iterates. Let z0 be an arbitrarily element of C , Construct a sequences { z n } ofpoints 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 thecomplex-valued function f with an initial choice z0 and parameter s . We may denote it 264
  • 3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEMEby SO ( f , x0 , sn ) . Notice that SO ( f , x0 , sn ) with sn = 1 is O ( f , x0 ) . We remark that thesuperior 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 solutionsproposed 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 PlaneDefinition 2.4. Phoenix set The Phoenix function was introduced by the Shigehiro Ushiki [12], using complexfunction 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 planedynamics of Phoenix function, generally known as Phoenix fractal, is a modification of theclassic Mandelbrot and Julia sets. The study of Ushiki shows that the phoenix set does nothave the same properties as the classical Mandelbrot Set see Fig. 2, represents the complexone-dimensional section of a “Julia-like” set of a complexified “Henon map”. Define a 2 2holomorphic automorphism of the two dimensional complex Euclidean space f : c → c by 265
  • 4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEMEf ( x, y) = ( x2 + c + by, x) , where c and b are the complex constants, see [12]. The phoenixappears when the attractive fixed point of this mapping loses its stability via a saddle-nodebifurcation. The parameter values are chosen as b = -0.5, c = 0.56667. The picture represents 2a 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 superiorsequence of iterates. We may denote it by SO ( f , xo , sn ) . Now we define the Mandelbrot setfor zn +1 = zn p + real (c) + img (c) zn−1 , where p ≥ 2 and n = 2, 3, 4,... with respect to Manniterates. The collection of points whose orbits are bounded under the superior iteration for thePhoenix 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 Phoenixfunction using superior iterates. Further, we have also presented the convergence of phoenixfunction for different values of s and c. For z0 = (-0.124, 1.61) and s = 0.3, we observe that thevalue for F (z) converge to a fixed point i.e. 0.77086, see Table 1 and Fig. 3. On increasingthe value to z0 = (-0.124, 1.61) and s = 0.5 we obtain two fixed points i.e. 0.4402 and 0.9908see Table 2 & Fig. 4. Further, on increasing the value to z0 = (-5.8347, 0.1359) and s = 0.4 wefind two fixed points i.e. 2.5896 and 1.2745 see Table 4 & Fig. 6. On increasing the value ofz0 to (-55, 0) and fixing at s to 0.14, we obtain two fixed points i.e. 4.5863 and 8.6995, seeTable 6 & Fig. 8. For z0 = (1.4246,-1.3085) and s = 0.3 we observe that the function escape toinfinity, see Table 3 & Fig. 5. For z0 = (0.4868, 1.1694) and s = 0.5 we find that the value isescape 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
  • 5. 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
  • 6. 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
  • 7. 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
  • 8. 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 thephoenix function. In most of the figures we found symmetry along x axis. As an exception wefound some Phoenix Julia sets symmetrical around x as well as y axis see Fig. 12. It isobserved that the orbit of Phoenix function converges to either 1 or 2 point. It is observed thatthe superior Julia sets for Phoenix function to be symmetric along x axis for even powers, seeFig. 15-16, and symmetric along y axis for the odd powers of p see Fig. 17-18. 270
  • 9. 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
  • 10. 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
  • 11. 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 = 35. CONCLUSION In this paper we have presented the dynamics and fixed point analysis of Phoenix setby using superior Iterates. Further we have presented the geometric properties of superiorJulia sets for Phoenix function along different axis. We have also presented an image whichresemble to a pair of leaf, see Fig. 11 and famous spider fractal see Fig. 10. Further, we havepresented the Phoenix fractals beyond the traditional values i.e. (2, 0), see Fig. 12 & 14.REFERENCES1. Barcellos, A. and Barnsley, Michael F., Reviews: Fractals Everywhere. Amer. Math. Monthly, No. 3, pp. 266-268, 1990.2. Barnsley, Michael F., Fractals Everywhere. Academic Press, INC, New York, 1993.3. Edgar, Gerald A., Classics on Fractals. Westview Press, 2004.4. Falconer, K., Techniques in fractal geometry. John Wiley & Sons, England, 1997.5. Falconer, K., Fractal Geometry Mathematical Foundations and Applications. John Wiley & Sons, England, 2003.6. Kumar, Manish. and Rani, Mamta., A new approach to superior Julia sets. J. nature. Phys. Sci, pp. 148-155, 2005.7. Negi, A., Fractal Generation and Applications, Ph.D Thesis, Department of Mathematics, Gurukula Kangri Vishwavidyalaya, Hardwar, 2006.8. Orsucci, Franco F. and Sala, N., Chaos and Complexity Research Compendium. Nova Science Publishers, Inc., New York, 2011. 273
  • 12. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME9. Peitgen, H. O., Jurgens, H. and Saupe, D., Chaos and Fractals. New frontiers of science, 1992.10. Peitgen, H.O., Jurgens, H. and Saupe, D., Chaos and Fractals: New Frontiers of Science. Springer-Verlag, New York, Inc, 2004.11. Rani, M., Iterative Procedures in Fractal and Chaos. Ph.D Thesis, Department of Computer Science. Gurukula Kangri Vishwavidyalaya, Hardwar, 2002.12. Ushiki, Shigehiro., Phoenix. IEEE Transaction on Circuits and System, Vol. 35, No. 7, pp. 788-789, 1998.13. Hitashi and Sugandha Sharma, “Fractal Image Compression Scheme Using Biogeography Based Optimization On Color Images” International journal of Computer Engineering & Technology (IJCET), Volume 3, Issue 2, 2012, pp. 35 - 46, Published by IAEME.14. Pardeep Singh, Nivedita and Sugandha Sharma, “A Comparative Study: Block Truncation Coding, Wavelet, Embedded Zerotree And Fractal Image Compression On Color Image” International journal of Electronics and Communication Engineering &Technology (IJECET), Volume 3, Issue 2, 2012, pp. 10 - 21, Published by IAEME. 274

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