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12Oriental Renaissance: Innovative, p VOLUME 2 | ISSUE 4
educational, natural and social sciences A ISSN 2181-1784
Scientific Journal Impact Factor Q SJIF 2022: 5.947
Advanced Sciences Index Factor ASI Factor = 1.7
IKKI O'LCHOVLI SIMPLEKSDA ANIQLANGAN KVAZI NOVOLTERRA KUBIK STOXASTIK OPERATORINING DINAMIKASI
Safarov Abbos Abdurasul o'g'li
"TIQXMMI" Milliy tadqiqot universitetmrng Qarshi irrigatsiya va agrotexnologiyalar instituti
ANNOTATSIYA
Ushbu maqolada matematikaning zamonaviy tatbiqlaridan biri novolterra kubik stoxastik operatorlarni kvazi sharti ostida ikki o'lchovli simpleksdagi dinamikasi o 'rganilgan. Shuningdek, kvazi novolterra kubik stoxastik operatorning qo 'zg'almas nuqtasining yagonaligi haqida teorema isbotlangan.
Kalit so izlar: kubik operator, kvazi, novolterra, qo 'zg 'almas nuqta, stoxastik.
ABSTRACT
The paper studies the dynamics of non-Volterra cubic stochastic operators in a two dimensional simplex under quasi conditions, one of the modern applications of mathematics. A theorem on the uniqueness of a fixed point of a quasi non-Volterra cubic stochastic operator is also proved.
Keywords: cubic operator, quasi, non-Volterra, fixed point, stochastic.
АННОТАЦИЯ
В статье исследуется динамике неволътерровых кубических стохастических операторов в двумерном симплексе при квазиугловиях, одным из современных приложений математики. Также доказана теорема о единственности неподвижной точки квази неволътеррова кубического стохастического оператора.
Ключевые слова: кубический оператор, квази, неволътерра, неподвижная точка, стохастический.
Quyidagi
S"1 =|х = (Xi,х2,...,xn)eD ": х >0,i = \П, Jx = 1
to'plam (n -1) o'lchovli simpleks deb ataladi [3]. W: Sn-1 ^ Sn-1 akslantirish,
n
(щ) = x = J PfftJXiXx,I = {1,2,...,n) (1)
i, j ,k=i
bu yerda:
Oriental Renaissance: Innovative, educational, natural and social sciences Scientific Journal Impact Factor Advanced Sciences Index Factor
VOLUME 2 | ISSUE 4 ISSN 2181-1784 SJIF 2022: 5.947 ASI Factor = 1.7
n
p = p = p = p = p = p > o v p = i
ijk, l jik ,l kji ,l kij ,l jki ,l ikj, l ? ijk ,l
(2)
l=i
(1), (2) ni kubik stoxastik operator deb ataymiz.
Agar P..kl = 0, V/ e{i, j,k) (3) shart o'rinli bo'lsa, (1), (2) operator novolterra
kubik stoxastik operatori deb ataladi [1], [4].
Agar novolterra operatorining faqat P. va P^ l, i ^ j ^ k koeffitsiyentlari uchun
(3) o'rinli bo'lmasa, W operator kvazi novolterra kubik stoxastik operator deb ataladi
[4].
Kvazi novolterra kubik stoxastik operatorni n = 3 da o'rganamiz: x' = ax x3 + Px y3 + y z3 + 3 y2 z + 3 yz2 + 2 xyz, y' = a2 x3 + P2 y3 + y2 z3 + 3x2 z + 3xz2 + 2 xyz, (4)
z' = a3 x3 + j33 y3 + y3 z3 + 3x2 y + 3xy2 + 2 xyz,
W:
3 3 3
Bu yerda a,ß,y > 0, i = 1,2,3, ^a = ^ß = ^y = 1.
a=y=a2= ß2 = ß = y3 = 0, a3 = ß = y2 = 1 bo'lgan holda o'rganamiz. Ushbu holda operator quyidagi ko'rinishga keladi: x' = y3 + 3 y2 z + 3 yz2 + 2 xyz W: <!y' = z3 + 3x2z + 3xz2 + 2xyz (5)
z' = x3 + 3x2 y + 3xy2 + 2 xyz (5) operatorning qo'zg'almas nuqtalarini W(A) = A, A = (x, y, z) tenglamani yechish orqali aniqlaymiz. Ya'ni
y3 + 3 y2 z + 3 yz2 + 2 xyz = x
z3 + 3x2 z + 3xz2 + 2 xyz = y (6)
x3 + 3x2 y + 3xy2 + 2 xyz = z Fix(W) orqali W operatorning barcha qo'zg'almas nuqtalari to'plamini belgilaymiz. Fix(W) = {Ae S2: W(A) = A}.
Quyidagi belgilashlarni kiritamiz: intS2 = {(x,y,z) e S2: xyz > 0},
'1 1 P
öS2 = S2 \ int S2 va C =
3'3'3
Teorema. (5) operator uchun quyidagilar o'rinli:
a) Fix(W) nöS2 =0
i=1
i=1
i=1
Oriental Renaissance: Innovative, p VOLUME 2 | ISSUE 4
educational, natural and social sciences ISSN 2181-1784
Scientific Journal Impact Factor Q SJIF 2022: 5.947
Advanced Sciences Index Factor ASI Factor = 1.7
b) Flx(W) n int S2 = {C}.
Isbot. a) (x,y,z) edS2 bo'lsin. Faraz qilaylik, x = 0 ( y = 0,z = 0 hollar ham xuddi shunday tekshiriladi) bo'lsin. U holda (6) sistemaga ko'ra, tenglamalar sistemasining yechimi x = y = z = 0 ekanligi kelib chiqadi. Lekin ta'rifga ko'ra
(0,0,0) £ S2. Bundan ko'rinadiki,
Flx(W) ndS2 =0
b) (x, y, z) e int S2 bo'lsin. (6) sistemaning birinchi va ikkinchi tenglamalarini ayiramiz:
y3 - z3 + 3y2 z - 3x2 z + 3yz2 - 3xz2 = x - y yoki
(y - z)( y2 + yz + z2) = (x - y )(1 + 3z (x + y) + 3z2) (7)
Xuddi shunday (6) sistemaning birinchi va uchinchi tenglamalarini ham ayiramiz:
y3 - x3 + 3y2z - 3x2y + 3yz2 - 3xy2 = x - z, (y - x)(y2 + xy + x2) = (x - z )(1 + 3y2 + 3y (x + z)) (8)
Ikkinchi va uchinchi tenglamalarning ham ayirmasini sodda holga keltiramiz: ( z - x)(z2 + zx + x2 ) = ( y - z )(l + 3x (y + z) + 3x2) (9)
V (x, y, z) e int S2 uchun,
y2 + yz + z2 > 0, y2 + xy + x2 > 0, z2 + zx + x2 > 0,
1 + 3z(x + y) + 3z2 > 0, 1 + 3y2 + 3y(x + z) > 0, 1 + 3x(y + z) + 3x2 > 0.
Faraz qilaylik z > x(z < x) bo'lsin. (7), (8) va (9) tenglamalardan
x > y (x < y) va y > z (y < z) ekanligi kelib chiqadi. Bundan esa
z > x > y > z (z < x < y < z) (10) munosabatga kelamiz. Shunday qilib, (6)
1 (111 ^ tenglamalar sistemasi yagona x = y = z = - yechimga ega. Demak, C = -,-,-
3 y 3 3 3 y
nuqta (5) operatorning yagona qo'zg'almas nuqtasi bo'ladi. Teorema isbotlandi.
REFERENCES
1. R.L. Devaney, An introduction to chaotic dynamical systems, stud. Nonlinearity, Westview Press, Boulder, CO 2003.
Oriental Renaissance: Innovative, educational, natural and social sciences Scientific Journal Impact Factor Advanced Sciences Index Factor
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R
VOLUME 2 | ISSUE 4 ISSN 2181-1784 SJIF 2022: 5.947 ASI Factor = 1.7
2. U.U.Jamilov, A.Yu.Khamraev, M.Ladra, On a Volterra cubic stochastic operator, Bull. Math. Biol. 80 (2) (2018) 319-334.
3. A. Yu. Khamraev, On cubic operators of volterra type (Russian), Uzbek. Math. Zh. 2004 (2) (2004) 79-84
4. U. A. Rozikov, A. Yu. Khamraev, On construction and a class of non-Volterra cubic stochastic operators, Nonlinear Dyn. Syst. Theory 14 )1) (2014) 92-100
