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The properties of the iterated complex Carotid–Kundalini function, given byzj+1=cosNzjcos−1zj+c,where z,c, and N are complex constants, are studied. Depending on the values of N,c, and z0 used, trajectories either tended to a fixed point, displayed periodic behaviour, or increased in value in an unbounded manner at each iteration. Trajectories that tended to fixed points were observed to do so in an arc-like manner for some parameter values. Trifurcations were seen as convergence to a single fixed point which was replaced by period-3 and period-9 motion, as the imaginary part of c was varied.
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E-mail address: 006grc@cosmos.wits.ac.za (G.R.J. Cooper).
Computers & Graphics 25 (2001) 153}158
Chaos and Graphics
Julia sets of the complex Carotid}Kundalini function
Gordon R.J. Cooper*
Department of Geophysics, University of the Witwatersrand, Private Bag 3, Johannesburg WITS 2050, South Africa
Abstract
The properties of the iterated complex Carotid}Kundalini function, given by
zH>"cos(NzHcos\(zH))#c,
where z,c, and Nare complex constants, are studied. Depending on the values of N,c, and zused, trajectories either
tended to a "xed point, displayed periodic behaviour, or increased in value in an unbounded manner at each iteration.
Trajectories that tended to "xed points were observed to do so in an arc-like manner for some parameter values.
Trifurcations were seen as convergence to a single "xed point which was replaced by period-3 and period-9 motion, as the
imaginary part of cwas varied. 2001 Published by Elsevier Science Ltd.
Keywords: Fractals
1. Introduction
The iterated equation
zH>"z
H#c, (1)
where zand care complex numbers and jis the iteration
number, has many fascinating mathematical properties
and can be used to produce a wide range of interesting
images [1, p. 791]. If, for a given value of cand a given
initial value z, the value of zHremains bounded after
repeated iterations, then zis said to lie within the Julia
set characterised by c. The boundary between those
values of zthat remain bounded after repeated iter-
ations and those that escape to in"nity is a fractal. Di!er-
ent fractals are formed when di!erent values of care used,
and they may consist either of only one piece or of many,
i.e. they may be connected or disconnected. The Mandel-
brot set is made up of all the values of cin the complex
plane for which the Julia set associated with that value is
connected [1, p. 843]. The Julia and Mandelbrot sets
based on Eq. (1) will be referred to in this paper as the
standard Julia and Mandelbrot sets.
Eq. (1) can be modi"ed in many di!erent ways to yield
Julia sets of di!erent kinds. This paper examines the
results of using the complex Carotid}Kundalini function
in place of the quadratic function used above.
2. The Carotid}Kundalini function
The Carotid}Kundalini (C}K) function is given by
[2, p. 179]
y(x)"cos(Nx cos\(x)). (2)
Eq. (2) can be modi"ed to put it into the form of (1):
zH>"cos(NzHcos\(zH))#c, (3)
where [3, p. 35]
cos z"(eGX#e\GX)/2 (4)
and [3 , p. 36]
cos\ z"1/iln(z#((z!1)). (5)
Eq. (3) has two parameters: the complex constant cand
the value N. In Eq. (2) Nis real, but here the equation is
generalised by allowing Nto be complex. A plot of it is
shown in Fig. 1 for two di!erent values of N.
0097-8493/01/$ - see front matter 2001 Published by Elsevier Science Ltd.
PII: S 0 0 9 7 - 8 4 9 3 ( 0 0 ) 0 0 1 1 4 - X
Fig. 1. Plot of the Carotid}Kundalini function for N"1 and 2.
Repeated iteration of Eq. (3) produced three di!erent
types of behaviour, depending on the initial value of
zchosen. The trajectories either increased in value with-
out limit at each iteration, tended to a "xed point, or
displayed periodic motion (see below). Fig. 2 shows the
arc-like paths displayed when trajectories were attracted
to the "xed point at 1.0#0.0i(when N"1.0#0.0iand
c"0.0#0.0i).
3. Julia sets
Fig. 3 shows the Julia set obtained by plotting the set
of the points which remained bounded after repeated
iterations of Eq. (3) (the prisoner set), for various values of
the real part of N(with cstill set to 0.0#0.0iand the
imaginary part of Nset to zero). As R(N) was increased,
the area occupied by the bounded trajectories decreased
154 G.R.J. Cooper /Computers & Graphics 25 (2001) 153}158
Fig. 2. Arc-like convergence trajectories observed (N"1.0#0.0iand c"0.0#0.0i).
and took on a &('shaped appearance. Fringes became
apparent around the main &('shape, becoming clearer
as R(N) increased. When R(N)"1.0 copies of "gures
similar to the standard Mandelbrot set appeared
throughout the "gure, as did dipolar patterns caused by
the curved paths taken by the bounded trajectories. For
the standard Julia set these dipolar patterns are termed
&petals'[4]. As R(N) was increased further, the area
occupied by the bounded trajectories decreased further,
and merged with the nearby fringes. A second layer of
fringes, di!erent in character to the "rst, then became
apparent. All calculations were performed using
Matlab 5.3.
Fig. 4 shows the Julia set obtained by plotting the
prisoner set of Eq. (3), for various values of the imaginary
part of N(with cstill set to 0.0#0.0iand the real part of
Nset to zero this time). It is similar (though not exactly
the same) to Fig. 3, only re#ected about the imaginary
axis, giving a &''shape. Fig. 5 shows the prisoner set
obtained when N"1.0#1.0i(c"0.0#0.0i) as this has
adi!erent character to the cases when N"1.0#0.0ior
N"0.0#1.0i. There is an outer pattern of fringes,
similar to that found when R(N) exceeded 1.5, inside of
which were a series of di!use semicircular arcs. Between
these outer arcs were smaller arcs, from which conver-
gence to a "xed point required more iterations than did
convergence from the outer arcs. This structure was then
repeated (apparently inde"nitely), with yet smaller arcs
being found.
Movie "les have been produced (in Matlab format)
that show a smooth animation of the change in the
prisoner set as both R(N) and I(N) were varied. They can
be downloaded via ftp from ftp.cs.wits.ac.za in directory
/pub/general/geophys.
Finally, the e!ect of varying cwas explored. When the
real part of cwas varied (with I(c)"0.0), the prisoner set
took on a very sparse appearance for all values tested.
The general form was a &('shape similar to those
produced when R(N) was varied. The same was found
when values of the imaginary part of cwere used, except
when I(c) was less than approximately 0.5 (with
R(c)"0.0), when a more &solid'plot was obtained.
Fig. 6 shows a plot of the "xed point against the
imaginary part of c, while the real part was held constant
at 0.0. As I(c) was increased from 0.0 to 0.54, the "xed
point moved smoothly to +0.647#0.46i, but when I(c)
G.R.J. Cooper /Computers & Graphics 25 (2001) 153}158 155
Fig. 3. The Julia set of the complex Carotid}Kundalini function for di!erent values of the real part of N.
became 0.55 it was replaced by period-3 motion through
the points 0.852#0.454i, 0.439#0.267i, and
0.602#0.686i. The three points formed a triangle whose
centre was approximately at 0.647#0.46i.AsI(c) was
increased further, the amplitude of the period-3 motion
increased, until at I(c)+0.67 period-9 motion appeared.
The number of iterations required for convergence to the
periodic values increased substantially as the trifurcation
156 G.R.J. Cooper /Computers & Graphics 25 (2001) 153}158
Fig. 4. The Julia set of the complex Carotid}Kundalini function for di!erent values of the imaginary part of N.
point was approached from below. The period-9 points
were close in value to the previous period-3 points, and
are indistinguishable from them on the scale of the plot in
Fig. 6. It was only possible to increase I(c) further by
a very small amount before the period-9 "xed points
became unstable and all trajectories diverged to in"nity.
G.R.J. Cooper /Computers & Graphics 25 (2001) 153}158 157
Fig. 5. The Julia set of the complex Carotid}Kundalini function
for N"1.0#1.0iand c"0.0#0.0i.
Fig. 6. Fixed points of complex Carotid}Kundalini function trajectories as I(c) is varied (R(c)"0 and N"1.0#0.0i).
4. Conclusions
The complex iterated Carotid}Kundalini function
produced a range of fractal Julia sets in the complex
plane as the parameters cand Nwere varied. Periodic
behaviour was observed in some trajectories that re-
mained bounded.
References
[1] Peitgen H-O, Jurgens H, Saupe D. Chaos and fractals, new
frontiers of science. New York: Springer, 1992 984pp.
[2] Pickover CA. Keys to in"nity. New York: Wiley, 1995.
[3] Speigel MR. Theory and problems of complex variables.
New York: McGraw-Hill, 1972, 313pp.
[4] Blanchard P. Complex analytic dynamics on the Riemann
sphere. Bull Amer Math Soc 1984;11(1):85}141.
158 G.R.J. Cooper /Computers & Graphics 25 (2001) 153}158
... In mathematics the Mandelbrot set, named after Benoit Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial 2 1 nn z z c   remains bounded see[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Julia sets [13] provide a most striking illustration of how an apparently simple process can lead to highly intricate sets. ...
... Where n= number of iterations, is iterated, many fascinating properties are revealed as given by Cooper[1][2].By using the above equation, Cooper produced a range of fractal Julia sets in the complex plane as parameters c and N were varied [2]. Also periodic behaviour was observed in some trajectories that remained bounded. ...
... Since then, numerous generalizations of fractals have come into the existence via different fixed point iterative methods (for instance, Mann-iteration [9], Ishikawa-iteration [10], S-iteration [11], Noor-iteration [12], Jungck-type iterations [13][14][15][16][17][18][19], CR-iteration [20], SP-iteration [21], and so on). Fractals for transcendental complex functions were studied in [22][23][24][25][26][27][28]. It is interesting to mention here that an iteration method, also known as a method of successive approximations, permits finding a solution with a predetermined accuracy and may be easily programmed. ...
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Theory and problems of complex variables
  • M R Speigel
Speigel MR. Theory and problems of complex variables. New York: McGraw-Hill, 1972, 313pp.