Definition 4.
The filled Julia set
c
x
f
K
n
,
of a mapping
c
x
f
n
,
is
defined as the set of all points
R
x
, that have bounded orbits
with respect to mapping
c
x
f
n
,
.
,
|
,
n
as
c
x
f
x
c
x
f
K
n
n
Definition 5.
The
Julia set
is the common boundary of the filled
Julia set
.
,
,
.
c
x
f
K
c
x
f
J
n
n
Definition 6.
The
Mandelbrot set
c
x
f
M
n
,
for the mapping
c
x
f
n
,
is the
set of all points
c
on the parameter plane (or line), which the orbits
of the all critical points are bounded.
125
Definition 7.
If the orbit have following three properties then it is
chaotic
:
i.
Dense periodic points.
ii.
Transitivity.
iii.
Sensitive dependence of initial condition.
1.
Algorithms for developing computer programs
First algorithm is for filled Julia set on Euclidean plane for the mapping
q
y
x
g
y
p
y
x
f
x
F
n
n
n
n
n
n
,
,
,
,
:
1
1
where p and q are parameters
Algorithm JS. (For filled Julia set)
x=xmin-step
while x< xmax {
y=ymin-step
x=x+step
while y< ymax {
y=y+step
k=0
x1=x
y1=y
while (x1*x1+y1*y1k=k+1
xm=x1
x1=f(x1,y1,p)
y1=g(x1,y1,q) }
if k=N then Print(x,y) } }
Second algorithm is for Mandelbrot set on Euclidean plane for the mapping
q
y
x
g
y
p
y
x
f
x
F
n
n
n
n
n
n
,
,
,
,
:
1
1
where p and q are parameters.
Algorithm MS. (For Mandelbrot set)
x=xmin-step
while x< xmax {
y=ymin-step
x=x+step
while y< ymax {
y=y+step
k=0
x1=x
y1=y
while (x1*x1+y1*y1k=k+1
xm=x1
x1=f(x1,y1,p)
y1=g(x1,y1,q) }
if k=N then Print(p,q) } }
2.
Examples. Example 1.
In this example we show filled Julia sets for following
mappings on
2
R
to itself
.
2
:
1
2
2
1
q
y
x
y
p
y
x
x
F
n
n
n
n
n
n
(1)
126
In our program we chosen R=6, N=50 xmin=-2, xmax=2, ymin=-2, ymax=2, step=0,0001.
If p=q=0 then filled Julia set is unit circle center on origin Fig 1.
If p=0,25 and q=0 then filled Julia set is called the “cauliflower” Fig 2 which example of
the fractal.
If p=-0,75 and q=0 then we get Fig 3.
When p=-0,1 and q=0.8 then our fractal is called Doudy’s rabbit every where two ears.
by name of American mathematics Andrean Doudy Fig 4.
When p= 0,360284 and q= 0,100376 then filled Julia set is in Fig 5.
The sets of all parameters (p,q) which corresponding filled Julia sets are connected is
Mandelbrot set Fig 6.
127
Example 2.
In this example we show filled Julia sets for following mappings on
2
R
to
itself
.
:
2
1
2
1
q
x
y
p
y
x
F
n
n
n
n
(2)
In this case filled Julia sets are regular rectangle for (p,q) in M fig 7.
And Mandelbrot set for (2) mapping is in fig 8.
Theorem 1.
There are exist on boundary orbits of the mapping (1) that they are chaotic.
Theorem 2.
If p=q=-2 then the orbits of the mapping (2) are chaotic.
Reference
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Boqiyev A.M. Uzluksiz ta’lim tizimida o‘quvchilarning algoritmik kompetentsiyalarini
shakllantirishda dasturlash imkoniyatlaridan foydalanish
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