Complex dynamics of superior phoenix set

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

6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 1, January- February (2013), © IAEME

263

COMPLEX DYNAMICS OF SUPERIOR PHOENIX SET

Sunil Shukla *, Ashish Negi **

* Department of Computer Science

Omkarananda Institute of Management & Technology, Rishikesh

Tehri Garhwal, 249192.

Email: [email protected]

** Department of Computer Science & Engineering

G.B Pant Engineering College, Pauri Garhwal, 246001.

Email: [email protected]

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 1 1( ) ( )p

n n nz z real c img c z

+ −= + + , 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

2

n nz z c= + give rise of fractals of an exotic appearance [1]. This function nz 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 2

1 1( ) ( )n n nz z r e a l c im g c z+ −

= + + , where n and c are constants.

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264

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 2

1 , ,n n

z z 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 0x A∈ , construct a

sequence { }nx in A in the following manner

( ) ( )

( ) ( )

( ) ( )

1 1 0 1 0

2 2 1 2 1

1 1

1

1

1n n n n n

x s f x s x

x s f x s x

x s f x s x− −

= + −

= + −

= + −

M

Where 0 1ns< ≤ and { }ns is convergent to a non-zero number.

The sequence { }nx constructed above is called Mann sequence of iterates or superior

sequence of iterates. Let 0z be an arbitrarily element of C , Construct a sequences { }nz of

points of C in the following manner:

( ) ( )1 1,1 1, 2,3....,n n n

z sf z s z n− −

= + − =

where f is a function on a subset of C and the parameter s lie in the closed interval[ ]0,1 .

The sequence{ }nz constructed above, denoted by ( )0, ,SO f z s is superior orbit for the

complex-valued function f with an initial choice 0z and parameter s. We may denote it



265

Fig. 1. Henon Map in real Plane

by ( )0, ,n

SO f x s . Notice that ( )0, ,n

SO f x s with 1ns = is ( )0,O f x . We remark that the

superior orbit reduces to the usual Picard orbit when 1ns = .

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. 2

1

1

1n n n

n n

x ax by

y x

+

+

= + +

=

The values used to produce chaotic solutions are a = -1.4, b = 0.3.

Definition 2.4. Phoenix set

The Phoenix function was introduced by the Shigehiro Ushiki [12], using complex

function 1 1( ) ( )

p

n n nz z rea l c im g c z+ −

= + + , 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

holomorphic automorphism of the two dimensional complex Euclidean space2 2:f c c→ by



266

2( , ) ( , )f x y x 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

a complex line { }x y= in

2c , 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 { }nx constructed above is called Mann sequence of iteration or superior

sequence of iterates. We may denote it by ( , , )o nSO f x s . Now we define the Mandelbrot set

for 1 1( ) ( )p

n n nz z real c img c z

+ −= + + , where 2p ≥ and 2,3, 4,...n = 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)



268


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




269

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




270


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.



271

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




272

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




273

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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