Complex dynamics of superior phoenix set

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
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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.

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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-

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


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

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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)

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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

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)

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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

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)

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

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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)

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

Fig. 7. F (z) for (z0 = 0.4868, 1.1694) at s = 0.5

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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

Fig. 8. F (z) for z0 = (-55, 0) at s = 0.14

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


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


271


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


272


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.

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Complex dynamics of superior phoenix set

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