One of the most actively developing directions of modern mathematics is the theory of dynamical systems. This theory studies the evolution of processes over time and is investigated in both discrete and continuous settings. Dynamical systems are widely applied in many fields such as biology, economics, physics, and information technology. Scientists such as Poincare, Lyapunov, and Kolmogorov played an important role in the formation of this theory.

The Uzbek mathematical school has also achieved significant results in this direction. In particular, stochastic operators, dynamical processes on simplices, and their stability properties have been deeply studied by T.A. Sarimsoqov, R.A. Ganikhodzhaev, N.N. Ganikhodzhaev, U.A. Rozikov, and U. Jamilov.

Let ( ) Sn−1 = x ∈ Rn : xi ≥ 0, i = 1, n, xi = 1

be the (n − 1)-dimensional simplex, and define an operator V : Sn−1 → Sn−1 by x′ k = Pij,kxixj, k = 1, . . . , n. (1)

i,j=1

Here,

Pij,k = Pji,k ≥ 0, Pij,k = 1. (2)

Definition 1. The operator defined by 1-2 is called a quadratic stochastic operator (QSO).

The condition Pij,k = 0, k / ∈ {i, j} (3) is called the Volterra condition, while Pij,k = 0, k ∈ {i, j} (4) is called the non-Volterra condition. If 3 holds, then the operator is called a Volterra QSO; if 4 holds, it is called a non-Volterra QSO.

Definition 2. Let V : Sn−1 → Sn−1 be a quadratic stochastic operator. A point x∗ ∈ Sn−1 is called a fixed point of V if V (x∗) = x∗.

We define a quadratic stochastic operator V : S2 → S2 as follows:

x′ k = Pij,kxixj, k = 1, 2,

i,j=1

x′ 3 = x3 = x3xj.

For this operator, we have x2 = 1 − x1 − x3, x1 = x, which yields the quadratic function f(x) = Ax2 + Bx + C. (6)

From the equality x′ 3 = x1x3 + x2x3 + x2 3, we obtain P33,3 = 1, P33,1 = P33,2 = 0, P23,3 = , P13,3 = , and P12,1 + P12,2 + P12,3 = 1.

Let us introduce the following notations:

x3 = R, P11,1 = a1, P12,1 = a2, P13,1 = a3, P22,1 = b1, P23,1 = b2, P33,1 = 0.

Then the quadratic function takes the form f(x) = (a1 − 2a2 + b1)x2 + 2 ((a2 − b1)(1 − R) + R(a3 − b2)) x + b1(1 − R)2 + 2b2R(1 − R).

Thus, A = a1 − 2a2 + b1, B = 2 ((a2 − b1)(1 − R) + R(a3 − b2)) , C = b1(1 − R)2 + 2b2R(1 − R), where A ∈ [−2, 2] \ {0}, B ∈ [−3, 3], C ∈ [0, 1].

We begin by determining the fixed points of the function f(x) = Ax2 + Bx + C.

Theorem 1. If ( C ≤ A, B = 1 − 2√AC, then the function f(x) has a unique fixed point

x = ∈ [0, 1].

Proof : Fixed points satisfy Ax2 + Bx + C = x, which is equivalent to Ax2 + (B − 1)x + C = 0.

A unique solution exists when the discriminant is zero:

∆ = (B − 1)2 − 4AC = 0.

Hence, B = 1 ± 2√AC.

The case B = 1 + 2√AC yields a negative solution, while B = 1 − 2√AC gives √AC x = ∈ [0, 1], which holds if and only if 0 ≤ C ≤ A. □

Theorem 2. If the coefficients satisfy   −2 ≤ A < 0, 2A + B < 1, A + B + C ≥ 1, then f(x) has a unique fixed point x1 = .

Theorem 3. Let 0 < A ≤ 2. Then the function f(x) has the fixed point

if  2A + B < 1, ( A + B + C ≤ 1, or   B > 1 − 2√AC.

B > 1 − 2√AC, Moreover, it has the fixed point 1 − B + (B − 1)2 − 4AC

if  B > 1 − 2√AC,

A + B + C ≥ 1.

Theorem 4. (1) If ∆ = 0, then the fixed point is a hyperbolic attracting fixed point.

(2) The fixed point x1 is hyperbolic attracting when ∆ < 4, and hyperbolic repelling when ∆ > 4.

(3) The fixed point x2 is a hyperbolic repelling fixed point.

Proof : i) Consider the case ∆ = 0. In this case, we use topological conjugacy. Let f(x) = Ax2 + Bx + C, g(x) = µx(1 − x), h(x) = mx + n.

We use the conjugacy relation h ◦ f = g ◦ h.

Then h ◦ f = mAx2 + mBx + (mC + n), g ◦ h = −µm2x2 + µm(1 − 2n)x + µ(n − n2).

Comparing coefficients, we obtain the system  −µm = A, µ(1 − 2n) = B,   2 µ(n − n ) = mC + n.

Solving this system yields µ = 1, m = −A, n = .

This implies that the fixed point is attracting.

ii) It is known that for the fixed point

x1 = ,

The derivative satisfies f ′(x1) = 2Ax1 + B = 1 − √∆.

Therefore, the fixed point x1 is hyperbolic attracting when ∆ < 4 and hyperbolic repelling when ∆ > 4.

iii) For the fixed point 1 − B + (B − 1)2 − 4AC x2 = , we have f ′(x2) = 2Ax2 + B = 1 + √∆.

Hence, this fixed point is a hyperbolic repeller. □

The work is supported by the applied Project "Dynamics and Applications of Cubic Stochastic Operators"No. AL-9224093956 of The Agency for Innovative Development under the Ministry of Higher Education, Science and Innovation of the Republic of Uzbekistan.

All authors have read and agreed to the published version of the manuscript.

The authors declare that no external funding or support was received for the research presented in this paper, including administrative, technical, or in-kind contributions. Data Availability All data supporting the reported findings in this research paper are provided within the manuscript.

The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study’s design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.