Oʻzbekiston respublikasi оliy va oʻrta maxsus

Definition 4.[2] A fixed point x^∗ is called hyperbolic if its Jacobian 𝐷_x𝑉(x^∗) has no eigenvalues on the unit circle.
Definition 5.[2] A hyperbolic fixed point x^∗ is called:

1. 
   attracting if all the eigenvalues of the Jacobian 𝐷_x𝑉(x^∗) are less than 1 in absolute value;
   
2. 
   repelling if all the eigenvalues of the Jacobian 𝐷_x𝑉(x^∗) are greater than 1 in absolute value;
   
3. 
   a saddle otherwise.
   

3. 
   Main result. Consider the following two strictly non-Volterra QSOs on the two- dimensional simplex
   

𝑉:
𝑥′₁ =
𝑥′₂ =
_𝗅𝑥′₃ =




𝑥₁
𝑥₁
+ 2𝑥₁𝑥₂,

^1𝑥^2+ ^1𝑥^2+ ^1𝑥^2 3 1 3 2 3 3
1 3	2	+ ^1𝑥^2 + ^1𝑥^2 3 2 3 3
1	2	+ ^1𝑥^2 + ^1𝑥^2 2 3
3		3		3

+ 2𝑥₂𝑥₃,
+ 2𝑥₃𝑥₁.

(3)

Lemma 1. The center x is a unique and attracting point of the QSO (3).
Proof. The equation 𝑉(x) = x has the form



1 ₂ 1 ₂ 1 ₂
𝑥₁ = ₃ 𝑥₁ + ₃ 𝑥₂ + ₃ 𝑥₃ + 2𝑥₁𝑥₂,



1 ₂ 1 ₂ 1 ₂

𝑥₂ = ₃ 𝑥₁ + ₃ 𝑥₂ + ₃ 𝑥₃ + 2𝑥₂𝑥₃,
(4)

1 2	+ ^1𝑥^2 + ^1𝑥^2 2 3
𝑥1 3	3		3

_𝗅𝑥₃ =
+ 2𝑥₃𝑥₁.

As before, a solution of the system (4) is a fixed point. It is known that the QSO (3) is a continuous operator and that the simplex over a finite set is compact and convex, so that by the Brouwer fixed-point theorem there is always at least one fixed point. We rewrite the system (4) in the form
𝑥₁ − 𝑥₂ = 2𝑥₂(𝑥₁ − 𝑥₃),

{𝑥₂ − 𝑥₃ = 2𝑥₃(𝑥₂ − 𝑥₁),
𝑥₂ − 𝑥₁ = 2𝑥₁(𝑥₃ − 𝑥₂).
(5)

Let x^∗ ∈ 𝑆² be a solution of the system (5). Assume that 𝑥^∗ ≥ 𝑥^∗. The rest is similar to this case.
1 2
Since 𝑥^∗ ≥ 𝑥^∗ from the first equation of the system (5) we get 𝑥^∗ ≥ 𝑥^∗. Using it from the second equation
1 2 1 3
of the system (5) we have 𝑥^∗ ≥ 𝑥^∗. Using it from the last equation we obtain that 𝑥^∗ ≥ 𝑥^∗ , that is
3 2 2 1

𝑥^∗ ≥ 𝑥^∗ ⇒ 𝑥^∗ ≥ 𝑥^∗ ⇒ 𝑥^∗ ≥ 𝑥^∗ ⇒ 𝑥^∗ ≥ 𝑥^∗ ⇒ 𝑥^∗ ≥ 𝑥^∗ ≥ 𝑥^∗ ≥ 𝑥^∗
∗ ∗ ∗ ¹

1 2 1 3
3 2 2 1
1 3 2
₁ ⇒ 𝑥₁ = 𝑥₂ = 𝑥₃ = ₃.

Therefore it follows that the center c = (1/3,1/3,1/3) is a unique fixed point.′
To find the type of the unique fixed point, using 𝑥₃ = 1 − 𝑥₁ − 𝑥₂, we rewrite the quadratic opera- tor (3) in the form:

_′ 4 ₂ 2 ₂ 4


4 2 1

_{𝑥₁ = − ₃ 𝑥₁ + ₃ 𝑥₂ − ₃ 𝑥₁𝑥₂ + ₃ 𝑥₁ − ₃ 𝑥₂ + ₃ ,
(6)

′ ² 2



4 ₂ 4



2 4 1

𝑥₂ = ₃ 𝑥₁ − ₃ 𝑥₂ − ₃ 𝑥₁𝑥₂ − ₃ 𝑥₁ + ₃ 𝑥₂ + ₃ .
where (𝑥₁, 𝑥₂) ∈ {(𝑥₁, 𝑥₂): 𝑥₁, 𝑥₂ ≥ 0,0 ≤ 𝑥₁ + 𝑥₂ ≤ 1}, and 𝑥₁, 𝑥₂ are the first two coordinates of the points of the simplex 𝑆².
One has that the partial derivations has the form

1
∂ 𝑥^𝘍 8
= −
∂𝑥₁ 3
𝑥₁ −
4 4
₃ 𝑥₂ + ₃ ,
∂𝑥^𝘍

1
=
∂𝑥₂
4
₃ 𝑥₂ −
4 2
₃ 𝑥₁ − ₃,

2
∂ 𝑥^𝘍 4
= −
∂𝑥₁ 3
𝑥₁ −
4 2
₃ 𝑥₂ − ₃ ,
∂𝑥^𝘍

2
=
∂𝑥₂
8
₃ 𝑥₂ −
4 4
₃ 𝑥₁ + ₃

and their values at the center 𝐜 are equal

1
∂ 𝑥^𝘍
(c) = 0,
∂𝑥₁
∂ 𝑥^𝘍 1

1
(c) = − ,
∂𝑥₂ 3
∂𝑥^𝘍 2

2
(c) = − ,
∂𝑥₁ 3
∂ 𝑥^𝘍

2
(c) = 0.
∂𝑥₂

Thus, the Jacobian matrix of the operator (6) at the center c has the form:
0 − ¹
3
𝐷_x𝑉(c) = (₋ ² ₀ )
3

Now let’s find the eigenvalues of the matrix 𝐷_x𝑉(c):

−𝜇 − ^1
3 √²

𝑑𝑒𝑡 (₋ ²
3
_−𝜇) = 0 ⇒ 𝜇_1,2 = ± ₃ ⇒ |𝜇_1,2| ≤ 1.

This shows that the center c = (1/3,1/3,1/3) is an attracting point. The proof of lemma completed.

Lemma 2. The function 𝜑(x) = |𝑥₁ − 𝑥₂| ⋅ |𝑥₂ − 𝑥₃| ⋅ |𝑥₃ − 𝑥₁| is a Lyapunov function for the operator (3).
Proof. For a x ∈ 𝑆² from (3) one has
𝜑⁽𝑉(x)⁾ = ^|2𝑥₁𝑥₂ − 2𝑥₂𝑥₃^| ⋅ ^|2𝑥₂𝑥₃ − 2𝑥₁𝑥₃^| ⋅ ^|2𝑥₁𝑥₃ − 2𝑥₁𝑥₂^|

3

3
= 8^|𝑥₂^| ⋅ ^|𝑥₁ − 𝑥₃^| ⋅ ^|𝑥₃^| ⋅ ^|𝑥₂ − 𝑥₁^| ⋅ ^|𝑥₁^| ⋅ ^|𝑥₃ − 𝑥₂^|

_{( )} 𝑥₁+𝑥₂+𝑥_{3 ( )}
² ( ) ( )

= 8𝑥₁ ⋅ 𝑥₂ ⋅ 𝑥₃ ⋅ 𝜑 x
≤ 8 (
) ⋅ 𝜑 x
3
= ( )
3
⋅ 𝜑 x
≤ 𝜑 x .

Thus the function 𝜑(x) is a decreasing Lyapunov function for the operator (3). The proof of the lemma completed.
Lemma 3. lim 𝑉^𝑛(x⁽⁰⁾) = c for any initial point x⁽⁰⁾ ∈ 𝑆².
𝑛→∞
Proof. By Lemma 2 the function 𝜑(x) is a decreasing Lyapunov function. So from the form of
QSO (3) for the 𝐱^(𝒏), 𝑛 = 0,1,2, … one has

3
_{𝜑 x}(𝑛+1) ²



(𝑛)
₂ (3𝑛+1)



(0)

(
Therefore it follows that
) ≤ ( )
3
⋅ 𝜑(x
) ≤ ( )
3
⋅ 𝜑(x ).

lim 𝜑(x^(𝑛)) = 0 ⇔ lim x^(𝑛) = c

𝑛→∞
where we have used that 𝜑(x⁽⁰⁾) is a bounded value. The proof of the lemma completed.
𝑛→∞

Theorem. a) The QSO (3) has a unique fixed point c = (1/3,1/3,1/3);

2. 
   The fixed point 𝐜 is an attracting point;
   
3. 
   For any x⁽⁰⁾ ∈ 𝑆², the trajectory {x^(𝑛)} tends to the fixed point 𝐜;
   
4. 
   The QSO (3) is a regular transformation.
   

Proof. Collecting together all three Lemmas we complete the proof of Theorem.


References:

1. 
   Bernstein S. (1942). The Annals of Math. Stat. 13, p. 53 ‒ 63.
   
2. 
   Devaney R.L. (2003). An introduction to chaotic dynamical systems (New York, Westview Press).
   

3. Ganikhodzhaev R.N. (1992). Sb.Math. 183, p. 489 ‒ 497.

4. 
   Jamilov U., Ladra M. (2016). Qual. Theory Dyn. Syst. 15, p. 257 ‒ 268.
   
5. 
   Jamilov U., Ladra M. (2020). Qual. Theory Dyn. Syst. 19, p. 95 ‒ 106.
   
6. 
   Lyubich Y.I. (1992). Mathematical Structures in Population Genetics (Berlin: Sprenger).
   
7. 
   Mamurov B.J., Rozikov U.A., Xudayarov S.S. (2020). Markov Pros. Relat.Fields 26, 915 ‒ 933.
   
8. 
   Rozikov U.A., Zhamilov U.U. (2008). Math. Notes. 83, p. 554 ‒ 559.
   

9. Zhamilov U.U., Rozikov U.A. (2009). Sb. Math. 200, p. 1339 ‒1351.

Абдуллаева Хулкар Рахматуллаевна (Республика Маънавият ва маърифат маркази
мустақил тадқиқотчи-изланувчиси) ЭПИСТЕМОЛОГИЯНИНГ АХЛОҚИЙ ЖИҲАТЛАРИ


Аннотация. Мақолада эпистемологик масалалар, инсон билимлари даражаси, илмий билим- лар ва фаннинг ривожланиши ахлоқий жиҳатлари юзасидан фикрлар тадқиқ этилган. Илмий би- лимлар коммертизациялашуви, олимнинг бурчи ва масъулияти, фаннинг сиёсатга хизмат қилиши қандай оқибатларга олиб келиши ҳақида фикр-мулоҳазалар билдирилган, илмий билимларнинг фан учун илмий билишнинг муҳимлиги, фундаментал фанларнинг зарурияти очиб берилган.
Аннотация. В статье рассматриваются такие вопросы как эпистемологические ценнос- ти, уровень человеческих знаний, научные знания и моральные аспекты развития науки. Были вы- сказаны мнения о коммерциализации научных знаний, долге и ответственности ученого, послед- ствиях служения науки политике, важности научных знаний для науки, раскрыта необходимость фундаментальных наук.
Annotation. The article deals with such issues as epistemological values, the level of human know- ledge, scientific knowledge and moral aspects of the development of science. Opinions were expressed on the commercialization of scientific knowledge, the duty and responsibility of a scientist, the consequences of serving science to politics, the importance of scientific knowledge for science, the need for fundamental sciences was revealed.
Калит сўзлар: эпистемология, фан коммертизациялашуви, сиёсат, ахлоқ ва қадрият, лойи- ҳа, ташқи ва ички омиллар, экстернализм, “салбий этика”.