Docstoc

COMPLEX DYNAMICS OF SUPERIOR PHOENIX SET

Document Sample
COMPLEX DYNAMICS OF SUPERIOR PHOENIX SET Powered By Docstoc
					  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. Further, we have
presented the Phoenix fractals beyond the traditional values i.e. (2, 0), see Fig. 12 & 14.

REFERENCES

1. 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
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME

9. 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

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:1
posted:2/21/2013
language:
pages:12