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4Oriental Renaissance: Innovative, educational, natural and social sciences Scientific Journal Impact Factor Advanced Sciences Index Factor
VOLUME 2 | ISSUE 12 ISSN 2181-1784 SJIF 2022: 5.947 ASI Factor = 1.7
КОМПЬЮТЕР ДАСТУРЛАРИ ЁРДАМИДА ФРАКТАЛ ФИГУРАЛАРНИ ТАСВИРЛАШНИ УРГАНИШ МЕТОДЛАРИ
Хозирги вацтда "хаотик ночизицли динамик системалар" математиканинг энг популяр йуналишларидан биридир. Реал уодисаларнинг аксарияти чизицли эмас. Ушбу мацолада биз фрактал фигураларни компютер дастурлаш орцали тасвирлашни урганамиз. Фракталлар чизицли булмаган уодисаларга цатъий боглицдир. Исаак Нютон даврида уар бир реал уодиса тартибли ёки тургун эканлиги уацидаги илмий царашлар мавжуд эди. Кейинчалик Пуанкаре [2] купгина реал уодисалар тургун эмас, яъни улар "тартибсиз" эканлигини исботлади. Биринчи марта фрактал фигуралар компютерда Бенуа Манделброт томонидан 1980 йилда IBM компаниясининг бир нечта дастурчилари билан кузатилган. Кейинчалик Манделъброт [4] туплами пайдо булди, бу фрактал, Жюлиа [1] тупламларининг энг мууим хоссалари ва регуляр, тартибсиз уодисага боглицдир. Ушбу мацолада биз фрактал фигураларни олиш ва уларни табиий уодисаларнинг айрим соуаларида цуллаш учун ишлаб чицилаётган компютер дастурини урганамиз.
Калит сузлар: Исаак Нютон, "хаотик ночизицли динамик системалар", "тартибсиз".
МЕТОДЫ ИССЛЕДОВАНИЯ МОДЕЛИРОВАНИЯ ФРАКТАЛЬНЫХ ФИГУР С ПОМОЩЬЮ КОМПЬЮТЕРНЫХ ПРОГРАММ
АННОТАЦИЯ
В настоящее время «хаотические нелинейные динамические системы» являются наиболее популярным разделом математического моделирования. Многие явления реальной жизни нелинейны. В настоящей статье мы исследуем моделирование фрактальных фигур с помощью компьютерного программирования. Фракталы находятся в строгой зависимости с нелинейными явлениями. Во времена Исаака Ньютона существовало научное мнение, что каждое реальное явление закономерно или стабильно. Позднее Пуанкаре [2] заметил, что многие из реальных явлений не являются
Нишонов Самад Нишонович
ТерДУ(Термиз давлат университети) катта-укитувчи
+998919061894
samad.nishonov@bk.ru
АННОТАЦИЯ
Oriental Renaissance: Innovative, educational, natural and social sciences Scientific Journal Impact Factor Advanced Sciences Index Factor
VOLUME 2 | ISSUE 12 ISSN 2181-1784 SJIF 2022: 5.947 ASI Factor = 1.7
регулярными, т. е. «хаотическими». Впервые фрактальные фигуры наблюдал на компьютере Бенуа Мандельброт с несколькими программистами в компании IBM в 1980 году. Позже появляется множество Мандельброта [4], являющееся фрактальным, важнейшим инструментом множеств Жюлиа [1] и строго зависит от нерегулярного явления. В данной работе мы изучаем развивающую компьютерную программу для получения фрактальных фигур и применения их к некоторой области природных явлений.
Ключевые слова: Исаака Ньютона, «хаотические нелинейные динамические системы», «хаотическими».
METHODS FOR STUDYING THE SIMULATIONS OF THE FRACTAL FIGURES WITH COMPUTER PROGRAMS ABSTRACT
At the present time "chaotic nonlinear dynamical systems " is the most popular branch of the mathematical modeling. Many of real life phenomena are nonlinear. In the present paper we investigate the simulations of fractal figures by computer programming. Fractals are strictly dependence with the nonlinear phenomena. There was scientific view that every real phenomenon is regular or stable at the time of Isaac Newton. Later Poincare [2] observed many of the real phenomena are not regular i.e. they are "chaotic". At first time fractal figures observed on the computer by Benoit Mandelbrot with several programmers of at the company IBM in 1980. Later appears the set of Mandelbrot [4] which is fractal, the most important tool of the sets of Julia [1] and strictly depends on the irregular phenomenon. In this paper we learn the developing computer program to obtain fractal figures and apply them to some area of natural phenomena.
Keywords: Isaac Newton, "chaotic nonlinear dynamical systems", "chaotic".
INTRODUCTION
Let x„+1 = f (xn, c) is the mapping on R to itself.
Definition 1. The set of points {xn | xn+1 = f (xn, c)} is called the orbit of x0 for f (xn, c) mapping.
Definition 2. If the set of points {xn | хи+1 = f (x„, c)} if consist only one point then x0 is called fixed point for f (xn, c) mapping.
Definition 3. A complex geometric pattern exhibiting self-similarity in that small details of its structure viewed at any scale repeat elements of the overall pattern.
Definition 4. The filled Julia set K(f (xn, c)) of a mapping f (xn, c) is defined as the set of all points x e R, that have bounded orbits with respect to mapping f (xn, c)
Oriental Renaissance: Innovative, p VOLUME 2 | ISSUE 12
educational, natural and social sciences fV ISSN 2181-1784
Scientific Journal Impact Factor Q SJIF 2022: 5.947
Advanced Sciences Index Factor ~ ASI Factor = 1.7
K(f (xn, c)) = {x | fn (x, c) — ^ as n — ro}
Definition 5. The Julia set is the common boundary of the filled Julia set
J (.f (xn, c)) = d(K (f (xn, c))) Definition 6. The Mandelbrot set M (f (xn, c)) for the mapping f (xn, c) is the set of all points c on the parameter plane (or line), which the orbits of the all critical points are bounded.
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
F.\xn+1 = f (xn , yn , p)
' I yn+i = g (xn, yn, q) 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*y1<R) and (k<N) {
k=k+1
xm=x1
x 1 =f(x 1,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
F\xn+1 = f (xn , yn , P)
'1 yn+i = g (xn, yn, q) where p and q are parameters.
Oriental Renaissance: Innovative, p VOLUME 2 | ISSUE 12
educational, natural and social sciences fV ISSN 2181-1784
Scientific Journal Impact Factor Q SJIF 2022: 5.947
Advanced Sciences Index Factor ~ ASI Factor = 1.7
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*y1<R) and (k<N) {
k=k+1
xm=x1
x 1 =f(x 1,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 R2 to itself
{2 2 Xn+1 = X2 " y2 + P (1)
yn+i = 2xnyn + q-
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.
y
Fig.
If p=0,25 and q=0 then filled Julia set is called the "cauliflower" Fig 2 which example of the fractal.
Oriental Renaissance: Innovative, educational, natural and social sciences Scientific Journal Impact Factor Advanced Sciences Index Factor
VOLUME 2 | ISSUE 12 ISSN 2181-1784 SJIF 2022: 5.947 ASI Factor = 1.7
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.
Oriental Renaissance: Innovative, educational, natural and social sciences Scientific Journal Impact Factor Advanced Sciences Index Factor
VOLUME 2 | ISSUE 12 ISSN 2181-1784 SJIF 2022: 5.947 ASI Factor = 1.7
The sets of all parameters (p,q) which corresponding filled Julia sets are connected is Mandelbrot set Fig 6.
Example 2.
In this example we show filled Julia sets for following mappings on R2 to itself
F :
x
«+i
= y« + P
(2)
l7„+i = xn + q-
In this case filled Julia sets are regular rectangle for (p,q) in M fig 7.
Oriental Renaissance: Innovative, educational, natural and social sciences Scientific Journal Impact Factor Advanced Sciences Index Factor
VOLUME 2 | ISSUE 12 ISSN 2181-1784 SJIF 2022: 5.947 ASI Factor = 1.7
Example 3. In this example we consider one dimensional logistic mapping is applicable for epidemic satiation.
-Vl (l - Xn ) (3)
The bifurcation diagram [2] for logistic mapping is in Fig 9.
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VOLUME 2 | ISSUE 12 ISSN 2181-1784 SJIF 2022: 5.947 ASI Factor = 1.7
We can find proof of these theorem from [1], [3].
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
1. Ganikhodzhaev R, N., Seytov Sh. J. Multidimensional case of the Von Neuman problem. Annual Review of Chaos Theory, Bifurcations and Dynamical SystemsVol. 9, (2020) 27-42.
2. Devaney R. L. An Introduction to Chaotic Dynamical Systems. New York, 1989. Westview Press. 362 p.
3. Li, T., and Yorke, J. Period Three imply Chaos. American Mathematical Monthly. Vol. 82, pp. 985-992 (1975).
4. Mandelbrot B. Fractals Aspects of Nonlinear dynamics. Annals of the New York Academy of science. Vol. 357, pp. 249-252 (1980).
