1 Introduction

S. Bernstein developed the concept of the quadratic stochastic operator (QSO) in 1924 [1]. Since then, QSOs have been intensively studied because they emerge in various models in physics [2,3], biology [1,4,5], economics and different branches of mathematics, such as graph theory and probability theory [6-9]. In a biological context, QSOs can be applied in the area of population genetics. QSOs can describe a generation-by-generation distribution when the distribution of the original starting generation is provided. We highlight how these operators can be used to interpret data in population genetics. We can see how these operators function when analogously looking at a biological population closed to reproduction with outside members. For that population, we can assume that each member of this closed biological group has one or more varying species-specific traits: \(\{1, ..., m\}\). Let \(x^{(0)} = (x_1^{(0)}, ..., x_m^{(0)})\) be a probability distribution of the species at an initial state and let the heredity coefficient \(p_{ij,k}\) be the conditional probability \(p(k \setminus i,j)\) that the \(i^{th}\) and \(j^{th}\)members of the species have reproduced successfully to produce a \(k^{th}\)individual. The first generation from this union \(x^{(1)} = (x_1^{(1)}, \dots, x_m^{(1)})\) can be calculated using the total probability

\[x_k^{(1)} = \sum_{i=1}^m p(k \setminus i, j) P(i, j), \quad k = \overline{1, m}\]

Given that no difference exists between the \(i^{th}\) and \(j^{th}\) members in any generation, the original progenitors i, j are independent, i.e. \(P(i, j) = P_i P_j\). This condition suggests that

\[x_k^{(1)} = \sum_{i,j=1}^m P_{ij,k} x_i^{(0)} x_j^{(0)}, \quad k = \overline{1, m}.\]

Consequently, the relation \(x^{(0)} \to x^{(1)}\) represents a mapping V, which is known as the evolution operator. Beginning with the selected initial state \(x^{(0)}\), the population iteratively develops to the first generation set \(x^{(1)} = V(x^{(0)})\) and then to the subsequent set \(x^{(2)} = V(x^{(1)}) = V(V(x^{(0)})) = V^{(2)}(x^{(0)})\) through time. Hence, the discrete dynamical system presents the population system evolution states as follows:

\[x^{(0)}, \quad x^{(1)} = V(x^{(0)}), \quad x^{(2)} = V^{(2)}(x^{(0)}), \quad \cdots\]

One of the main issues that underlie this theory is finding the limit points of V for an arbitrary starting point \(x^{(0)}\). Studying the limit points of QSOs is a complicated task even in 2D simplex and this problem has not yet been solved. Numerous researchers have presented a specific class of QSOs and have examined their behavior, e.g. F-QSO [10], Volterra-QSO [11-14], permutated Volterra-QSO [15,16], \(\ell\)-Volterra QSO [17,18], Quasi-Volterra-QSO [19], non-Volterra-QSO [20,21], strictly non-Volterra QSO [22], non-Volterra operators, and others, produced via measurements [23-25]. Nevertheless, collectively, these classes cannot represent all the QSOs as a set. An attempt was made to study the behavior of nonlinear operators, which is regarded as the main problem in nonlinear operators. However, this problem has not been comprehensively studied because it relies upon a specified cubic matrix, \((P_{ijk})_{i,j,k=1}^{m}\) [26].

Recently, Ganikhodzhaev, et al. [27] introduced \(\xi^{(as)}\)-QSO, which is a new class of QSOs that depend on a partition of the coupled index sets (which have couple traits) \(P_m = \{(i,j): i < j\} \subset I \times I\) and \(\Delta_m = \{(i,i): i \in N\} \subset I \times I\). In the case of 2D simplex (m=3), \(P_3\) and \(\Delta_3\) have five possible partitions.

In [28,29], the \(\xi^{(s)}\)-QSO related to \(|\xi_1|=2\) of \(P_3\) with a point partition of \(\Delta_3\) was investigated (see Section 2). In [30,31] the \(\xi^{(a)}\)-QSO related to \(|\xi_1|=2\) of \(P_3\) with a trivial partition of \(\Delta_3\) was studied (see Section 2). The \(\xi^{(as)}\)-QSO related to \(|\xi_1|=3\) of \(P_3\) with a point partition of \(\Delta_3\) was examined by

Mukhamedov and Jamal [27,32]. Furthermore, \(\xi^{(s)}\)-QSO and \(\xi^{(a)}\)-QSO are related to \(|\xi_1|=1\) of \(P_3\) with point and trivial partitions of \(\Delta_3\), respectively, as discussed by Alsarayreh, et al. [33]. This indicates that all partitions of \(P_3\) were investigated with respect to the point and trivial partitions of \(\Delta_3\). Therefore, the main motivation for this study was to introduce new partitions of \(\Delta_3\). The current study classifies the operators generated by \(\xi^{(as)}\)-QSO with cardinality \(|\xi_i|=2\) of \(P_3\) and \(|\xi_i|=2\) of \(P_3\). To demonstrate this in the current report, Section 2 establishes a number of preliminary definitions. Section 3 presents the description and classification of \(\xi^{(as)}\)-QSOs. Section 4 explains how this study examined the behavior of \(V_3\) and \(V_{15}\), obtained from classes \(G_3\) and \(G_9\), respectively. Section 5 examines the behavior of \(V_{26}\) and \(V_{25}\), obtained from classes \(G_{13}\) and \(G_{14}\), respectively.

2 Preliminary

Several basic concepts need to be operationalized to ensure clarity of terms.

Definition: QSO is a mapping of the simplex

\[S^{m-1} = \left\{ x = (x_1, \dots, x_m) \in \square^m : \sum_{i=1}^m x_i = 1, \ x_i \ge 0, \ i = \overline{1, m} \right\}\] (1)

into itself with the form

\[x'_{k} = \sum_{i,j=1}^{m} P_{ij,k} x_{i} x_{j}, \quad k = \overline{1, m},\] (2)

where \(V(x) = x' = (x'_1, \dots, x'_m)\), and \(P_{ij,k}\) is a coefficient of heredity that satisfies the following conditions:

\[P_{ij,k} \ge 0, \quad P_{ij,k} = P_{ji,k}, \quad \sum_{k=1}^{m} P_{ij,k} = 1.\] (3)

Based on the preceding definition, each QSO \(V: S^{m-1} \to S^{m-1}\) can be uniquely defined by a cubic matrix \(\mathcal{P} = \left(P_{ijk}\right)_{i,j,k=1}^m\) with conditions Eq. (1) and Eq. (2). For \(V: S^{m-1} \to S^{m-1}\), we specify the set of fixed points as \(F_{ix}(V)\). Moreover, for \(x^{(0)} \in S^{m-1}\), we indicate the set of limiting points as \(\omega_V(x^{(0)})\). It must be remembered that Volterra-QSO is defined by Eq. (2) and condition in Eq. (3), with the additional assumption

\[P_{ii,k} = 0 \quad \text{if} \quad k \notin \{i, j\}. \tag{4}\]

The biological rule of Condition Vol is well-established: an offspring replicates the genotype (trait) of its parent(s). Volterra-QSO exhibits the following form:

\[x'_{k} = x_{k} \left( 1 + \sum_{i=1}^{m} a_{ki} x_{i} \right), \ k \in I,\] (5)

where

\[a_{ki} = 2P_{ik,k} - 1 \text{ for } i \neq k \ a_{ii} = 0, i \in I \ a_{ki} = -a_{ik} \text{ and } |a_{ki}| \leq 1.\] (6)

Remarkably, this kind of operator has received extensive research attention [11-13]. The concept of \(\ell\)-Volterra-QSO was first presented in [17] and formulated as follows.

Let \(\ell \in I\) be fixed. Suppose that the genetic heredity coefficient \(\{P_{ij,k}\}\) satisfies

\[P_{ii k} = 0 \text{ if } k \notin \{i, j\} \text{ for any } k \in \{1, ..., \ell\}, i, j \in I,\] (7)

\[P_{i_0,i_0,k} > 0 \text{ for some } (i_0, j_0), i_0 \neq k, j_0 \neq k, k \in \{\ell + 1, ..., m\}.\] (8)

Therefore, the QSO specified by Eq. (2), additional condition in Eq. (3), Eq. (7) and Eq. (8) is called \(\ell\) -Volterra-QSO.

Remark 1. The following points must be reinforced:

• 1. An \(\ell\)-Volterra-QSO is a Volterra-QSO if and only if \(\ell = m\).
• No periodic trajectory exists for Volterra-QSO [11]. However, such trajectories exist for ℓ-Volterra-QSO [17].

Following [27], each element \(x \in S^{m-1}\) is a probability distribution of set \(I = \{1, ..., m\}\). Let \(x = (x_1, \cdots, x_m)\) and \(y = (y_1, \cdots, y_m)\) be vectors obtained from \(S^{m-1}\). We say that x is equivalent to y if \(x_k = 0 \iff y_k = 0\). We denote this relation as \(x \sim y\). Let \(supp(x) = \{i : x_i \neq 0\}\) be a support of \(x \in S^{m-1}\). We say that x is singular to y and specify this relation as \(x \perp y\) if \(supp(x) \cap supp(y) = \emptyset\). Notably, if \(x, y \in S^{m-1}\), then \(x \perp y\) if and only if (x, y) = 0, where \((\cdot, \cdot)\) denotes a standard inner product in \(\square^m\). Of further note is that \(|\xi_i|\) indicates the cardinality of potential partitions of \(P_m\) and \(\Delta_m\). If the cardinal of partition of \(\Delta_m\) is the maximum or minimum, it is named the point partition or trivial partition, respectively. We formulate sets of coupled indexes as follows:

\[P_m = \{(i, j) : i < j\} \subset I \times I, \quad \Delta_m = \{(i, i) : i \in I\} \subset I \times I\]

For any specified pair \((i,j) \in P_m \cup \Delta_m\), we set a vector \(\mathbb{P}_{ij} = (P_{ij,1}, \dots, P_{ij,m})\). Evidently, \(\mathbb{P}_{ij} \in S^{m-1}\) because of condition in Eq. (3). In this case, let \(\xi_1 = \{A_i\}_{i=1}^N\) and \(\xi_2 = \{B_i\}_{i=1}^M\) be fixed partitions of \(P_m\) and \(\Delta_m\), giving: \(A_i \cap A_j = \emptyset\), \(B_i \cap B_j = \emptyset\), \(\bigcup_{i=1}^N A_i = P_m\), \(\bigcup_{i=1}^M B_i = \Delta_m\), where \(N, M \leq m\).

Definition: [27] QSO \(V: S^{m-1} \to S^{m-1}\) given by Eq. (2) and condition in Eq. (3) is considered a \(\xi^{(as)}\)-QSO with respect to partitions \(\xi_1\) and \(\xi_2\) if the following conditions are satisfied:

• 1. Let \(k \in \{1, ..., N\}\) and \((i, j), (u, v) \in A_k\), then \(\mathbb{P}_{ii} \sim \mathbb{P}_{uv}\) is considered.
• 2. Let \(k \neq \ell, k, \ell \in \{1, ..., N\}\), \((i, j) \in A_k\) and \((u, v) \in A_\ell\), then \(\mathbb{P}_{ij} \perp \mathbb{P}_{uv}\) is considered.
• 3. Let \(d \in \{1, ..., M\}\) and \((i, i), (j, j) \in B_d\), then \(\mathbb{P}_{ii} \sim \mathbb{P}_{ii}\) is considered.
• 4. Let \(s \neq h\), \(s,h \in \{1,...,M\}\) and \((u,u) \in B_s\) and \((v,v) \in B_h\), then \(\mathbb{P}_{uu} \perp \mathbb{P}_{vv}\) is considered.

Example 1. Let \(\xi := \{\{(2,3)\}, \{(1,2),(1,3)\}\}\) and \(\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\) be two possible partitions of \(P_3\) and \(\Delta_3\) respectively. It is observable that \(P_{12} \sim P_{13}\), \(P_{23} \perp (P_{12}, P_{13})\) and \(P_{11} \perp P_{22} \perp P_{33}\). Therefore, based on such facts, the following values for \(\mathbb{P}_{ij}\) and \(\mathbb{P}_{ii}\) can be considered: \(\mathbb{P}_{12} = (\alpha, 1 - \alpha, 0), \quad \mathbb{P}_{13} = (\alpha, 1 - \alpha, 0), \quad \mathbb{P}_{23} = (0, 0, 1) \quad \mathbb{P}_{11} = (1, 0, 0), \\ \mathbb{P}_{22} = (0, 1, 0), \quad \mathbb{P}_{22} = (0, 0, 1)\). Due to Eq. (2), one can calculate the following:

\[V := \begin{cases} x' = (x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2\alpha x^{(0)}z^{(0)} \\ y' = (y^{(0)})^2 + 2(1-\alpha)(x^{(0)})(1-(x^{(0)})) \\ z' = (z^{(0)})^2 + 2x^{(0)}z^{(0)} \end{cases}\]

Hence, the operator V is considered \(\xi^{(as)}\)-QSO.

3 Classification of \(\xi^{(as)}\) - QSO

This section presents the classification of \(\xi^{(as)}\)-QSOs on 2-dimensional simplex, i.e. m=3 and the cardinality of the potential partitions of \(P_3\) and \(\Delta_3\) is equal to 2. The potential partitions of \(P_3\) are listed as follows:

\[\begin{split} \xi_1 &\coloneqq \{\{(1,2)\}, \{(1,3)\}, \{(2,3)\}\}, |\ \xi_1 \models 3, \\ \xi_2 &\coloneqq \{\{(2,3)\}, \{(1,2), (1,3)\}\}, |\ \xi_2 \models 2, \\ \xi_3 &\coloneqq \{\{(1,3)\}, \{(1,2), (2,3)\}\}, |\ \xi_3 \models 2, \\ \xi_4 &\coloneqq \{\{(1,2)\}, \{(1,3), (2,3)\}\}, |\ \xi_4 \models 2, \\ \xi_5 &\coloneqq \{(1,2), (1,3), (2,3)\}, |\ \xi_5 \models 1. \end{split}\]

The potential partitions of \(\Delta_3\) are listed as follows:

\[\begin{split} &\xi_1 \coloneqq \{\{(1,1)\}, \{(2,2)\}, \{(3,3)\}\}, |\, \xi_1 \mid = 3, \\ &\xi_2 \coloneqq \{(1,1), (2,2), (3,3)\}, |\, \xi_2 \mid = 1, \\ &\xi_3 \coloneqq \{\{(1,1)\}, \{(2,2), (3,3)\}\}, |\, \xi_3 \mid = 2, \\ &\xi_4 \coloneqq \{\{(3,3)\}, \{(1,1), (2,2)\}\}, |\, \xi_4 \mid = 2, \\ &\xi_5 \coloneqq \{\{(2,2)\}, \{(1,1), (3,3)\}\}, |\, \xi_5 \mid = 2. \end{split}\]

Proposition 1. For a class of \(\xi^{(as)}\)-QSO generated from the possible partitions of \(P_3\) and \(\Delta_3\) with cardinality equal to 2, we determine the following:

• 1. The class of all \(\xi^{(as)}\)-QSOs that correspond to partition \(\xi_3\) of \(P_3\) and partition \(\xi_5\) of \(\Delta_3\) is conjugate to the class of all \(\xi^{(as)}\)-QSOs that corresponds to partition \(\xi_2\) of \(P_3\) and partition \(\xi_3\) of \(\Delta_3\).
• 2. The class of all \(\xi^{(as)}\)-QSOs that corresponds to the partition \(\xi_4\) of \(P_3\) and partition \(\xi_4\) of \(\Delta_3\) is conjugate to the class of all \(\xi^{(as)}\)-QSOs that corresponds to partition \(\xi_2\) of \(P_3\) and partition \(\xi_3\) of \(\Delta_3\).

Proof.

1. Under the general form of QSO given by Eq. (2) and Eq. (3), the coefficients \(\left(P_{ij,k}\right)_{i,j,k=1}^{3}\) of operator V of \(\xi^{(as)}\)-QSOs that correspond to partition \(\xi_5 = \{\{(2,2)\}, \{(1,1),(3,3)\}\}\) of \(\Delta_3\) and partition \(\xi_3 = \{\{(1,3)\}, \{(1,2),(2,3)\}\}\) of \(P_3\), satisfying the following conditions: i. \(\mathbb{P}_{11} \sim \mathbb{P}_{33}\) and \(\mathbb{P}_{22} \perp \mathbb{P}_{mm}\), m = 1,3; ii. \(\mathbb{P}_{12} \sim \mathbb{P}_{23}\) and \(\mathbb{P}_{13} \perp (\mathbb{P}_{12}, \mathbb{P}_{23})\), where \(\mathbb{P}_{ij} = (p_{ij,1}, p_{ij,2}, p_{ij,3})\). To perform \(V_\pi = \pi V \pi^{-1}\) transformation on operator V, where permutation \(\pi = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{pmatrix}\) and where \(V_\pi : x_k' = \sum_{i,j=1}^3 P_{ij,k}^\pi x_i x_j\), \(k = \overline{1,3}\), such that \(\mathbb{P}_{ij,k}^\pi = P_{\pi(i)\pi(j),\pi(k)}\), for any \(i,j,k=\overline{1,3}\). Equivalently, \(\mathbb{P}_{ij}^\pi = \pi \mathbb{P}_{\pi(i)\pi(j)}\) (in vector form) for any i,j=1,2,3. Subsequently, operator \(V_\pi\) corresponding to partitions \(\xi_3\) of \(\lambda_3\) and \(\xi_2\) of \(\lambda_3\) is presented by applying the permutation \(\pi\) for the coefficient of V that corresponds to partition \(\xi_5\) of \(\lambda_3\) and \(\xi_3\) of \(\lambda_3\). From this, the following relations are derived:

• a. \(\mathbb{P}_{11} \sim \mathbb{P}_{33}\) and \(\mathbb{P}_{22} \perp (\mathbb{P}_{11}, \mathbb{P}_{33})\). By applying the permutation \(\pi\) on \(\mathbb{P}_{11}^{\pi} = \mathbb{P}_{33}\), \(\mathbb{P}_{22}^{\pi} = \mathbb{P}_{11}\), and \(\mathbb{P}_{33}^{\pi} = \mathbb{P}_{22}\), we obtain \(\mathbb{P}_{33} \sim \mathbb{P}_{22}\) and \(\mathbb{P}_{11} \perp (\mathbb{P}_{22}, \mathbb{P}_{33})\). Therefore, the properties of \(\xi_3\) of \(\Delta_3\) are verified.
• b. \(\mathbb{P}_{12} \sim \mathbb{P}_{23}\) and \(\mathbb{P}_{13} \perp (\mathbb{P}_{12}, \mathbb{P}_{23})\). By applying the permutation \(\pi\) on \(\mathbb{P}_{12}^{\pi} = \mathbb{P}_{13}\), \(\mathbb{P}_{13}^{\pi} = \mathbb{P}_{23}\), and \(\mathbb{P}_{23}^{\pi} = \mathbb{P}_{12}\), we obtain \(\mathbb{P}_{12} \sim \mathbb{P}_{13}\) and \(\mathbb{P}_{23} \perp (\mathbb{P}_{12}, \mathbb{P}_{13})\). Therefore, the properties of \(\xi_2\) of \(P_3\) are verified.
• 2. Similarly, the second part of this theorem can be proven by choosing the following permutation: \(\pi = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}\). This process completes the proof.

The preceding discussion reveals that any \(\xi^{(as)}\)-QSO obtained from the class that corresponds with partitions \(\xi_5\) of \(\Delta_3\) and \(\xi_3\) of \(P_3\) or \(\xi_4\) of \(\Delta_3\) and \(\xi_4\) of \(P_3\) is conjugate to certain \(\xi^{(as)}\)-QSOs obtained from the class that corresponds to partitions \(\xi_3\) of \(\Delta_3\) and \(\xi_2\) of \(P_3\). To investigate the operators of class \(\xi^{(as)}\)-QSO that correspond to partitions \(\xi_2\) of \(P_3\) and \(\xi_3\) of \(\Delta_3\), the coefficient \(\left(P_{ij,k}\right)_{i,j,k=1}^3\) in special form is selected, as shown in the Tables 1 and 2.

Table 1 The possible values of \(P_{ii}\).

where \(\alpha, \beta \in [0,1][0,1]\) such that \(\alpha + \beta = 1\).

Table 2 The possible values of \(P_{ij}\).

The choices for Cases \((I_j, II_i)\), where i, j = 1, ..., 6, provide 36 operators. These operators can be defined accordingly:

\[\begin{split} V_1 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \\ y' = \beta(x^{(0)})^2 \\ z' = (y^{(0)})^2 + (z^{(0)})^2 + 2y^{(0)}z^{(0)} \end{cases} & V_2 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \\ y' = \beta(x^{(0)})^2 + 2y^{(0)}z^{(0)} \end{cases} & V_2 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 + 2y^{(0)}z^{(0)} \\ z' = (y^{(0)})^2 + (z^{(0)})^2 \end{cases} & V_3 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \\ y' = \beta(x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \end{cases} & V_4 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 \\ y' = \beta(x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \\ z' = (y^{(0)})^2 + (z^{(0)})^2 + 2y^{(0)}z^{(0)} \end{cases} & V_4 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 \\ y' = \beta(x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \end{cases} \\ V_5 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 \\ y' = \beta(x^{(0)})^2 + 2y^{(0)}z^{(0)} \end{cases} & V_6 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 + 2y^{(0)}z^{(0)} \\ z' = (y^{(0)})^2 + (z^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \end{cases} \\ V_7 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \end{cases} & V_8 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \\ z' = (y^{(0)})^2 + (z^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \end{cases} \\ V_8 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \\ z' = (y^{(0)})^2 + (z^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \end{cases} \\ V_8 &\coloneqq \begin{cases} x' = \alpha(x^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \\ z' = (y^{(0)})^2 + (z^{(0)})^2 + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)} \end{cases} \end{aligned}\]

\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]

\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]

As can be seen, this class of \(\xi^{(as)}\)-QSOs contains 36 operators. These operators are too numerous to explore individually. Thus, two operators \(V_a\) and \(V_b\) are considered (topologically or linearly) conjugate when a permutation matrix P such that \(P^{-1}V_aP = V_b\) exists. Let \(\pi\) be a permutation of set \(I = \{1, ...m\}\). For any vector x, we define \(\pi(x) = (x_{\pi(1)}, ..., x_{\pi(m)})\). Verifying if \(\pi\) is a permutation of set I corresponding to the given permutation matrix P is simple. When true, then \(P_x = \pi(x)\). In this case, two operators \(V_a\) and \(V_b\) are conjugate if and only if \(\pi^{-1}V_a\pi = V_b\) for any permutation \(\pi\). Therefore, one may classify such operators into small classes and examine only the operators within these classes.

Theorem 1. Let \(\{V_1, \dots, V_{36}\}\) be the \(\xi^{(as)}\)-QSO presented above. Then, these operators are divided into 18 non-isomorphic classes:

\[\begin{array}{lll} G_1 = \{V_1, V_8\} & G_2 = \{V_2, V_7\} & G_3 = \{V_2, V_7\} & G_4 = \{V_4, V_{12}\} \\ G_5 = \{V_5, V_9\} & G_6 = \{V_6, V_{10}\} & G_7 = \{V_{13}, V_{20}\} & G_8 = \{V_{14}, V_{19}\} \\ G_9 = \{V_{15}, V_{23}\} & G_{10} = \{V_{16}, V_{24}\} & G_{11} = \{V_{17}, V_{21}\} & G_{12} = \{V_{18}, V_{22}\} \\ G_{13} = \{V_{25}, V_{32}\} & G_{14} = \{V_{26}, V_{31}\} & G_{15} = \{V_{27}, V_{35}\} & G_{16} = \{V_{28}, V_{36}\} \\ G_{17} = \{V_{29}, V_{33}\} & G_{18} = \{V_{30}, V_{34}\}. \end{array}\]

Proof. Evidently, the partitions \(\xi_2\) of \(P_3\) and \(\xi_3\) of \(\Delta_3\) are invariant only under the permutation \(\pi = \begin{pmatrix} x^{(0)} & y^{(0)} & z^{(0)} \\ x^{(0)} & z^{(0)} & y^{(0)} \end{pmatrix}\). Therefore, the given operators should be classified in relation with the remuneration of their coordinates. Consequently, we must perform the \(\pi V \pi^{-1}\) transformation on all the operators.

\[V_{1}(\pi^{-1}(x^{(0)}, y^{(0)}, z^{(0)})) = V_{1}(x^{(0)}, z^{(0)}, y^{(0)}) =\] \[(\alpha(x^{(0)})^{2} + 2x^{(0)}y^{(0)} + 2x^{(0)}z^{(0)}, \beta(x^{(0)})^{2},\] \[(y^{(0)})^{2} + (z^{(0)})^{2} + 2y^{(0)}z^{(0)})\]

Starting with \(V_1\) as the first operator, we obtain the following:

Thus,

\[\pi V_1 \pi^{-1} = \left( \alpha (x^{(0)})^2 + 2x^{(0)} y^{(0)} + 2x^{(0)} z^{(0)}, (y^{(0)})^2 + (z^{(0)})^2 + 2y^{(0)} z^{(0)}, (\alpha - 1) \beta (x^{(0)})^2 \right)\] \[= V_8.\]

We can derive the other classes by following the same procedure. The proof is completed using this process.

4 Dynamics of classes \(G_3\) and \(G_9\)

This section explores the dynamics of \(\xi^{(as)}\)-QSO \(V_{3,15}: S^2 \to S^2\) selected from \(G_3\) and \(G_9\). Firstly, \(V_3\) is rewritten as follows:

\[V_{3} := \begin{cases} x' = \alpha (x^{(0)})^{2} + 2y^{(0)}z^{(0)} \\ y' = (1 - \alpha)(x^{(0)})^{2} + 2x^{(0)}(1 - x^{(0)}) \\ z' = (z^{(0)})^{2} + (y^{(0)})^{2} \end{cases}\](9)

The operator \(V_3\) can be redrafted as a convex combination \(V_3 = \alpha W_1 + (1 - \alpha)W_2\), where

\[W_{1} := \begin{cases} x' = (x^{(0)})^{2} + 2y^{(0)}z^{(0)} \\ y' = 2x^{(0)}(1 - x^{(0)}) \\ z' = (z^{(0)})^{2} + (y^{(0)})^{2} \end{cases}\](10)

and

\[W_{2} := \begin{cases} x' = 2y^{(0)}z^{(0)} \\ y' = 2x^{(0)} - (x^{(0)})^{2} \\ z' = (z^{(0)})^{2} + (y^{(0)})^{2} \end{cases}\](11)

The following regions are introduced and the behavior of \(\{W_1^{(n)}\}_{n=1}^{\infty}\) over them is investigated:

\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]

Proposition 2. The following statements hold for \(W_1\) of \(V_3\):

• 1. The region \(A_1\) is invariant.
• 2. Let \(x_1^{(0)} \notin Fix(W_1)\), and \(x_1 \in A_2\), then, \(\{W_1^{(n)}\}_{n=1}^{\infty}\) goes to \(A_1 \cup A_3\). Accordingly, if \(x_1 \in A_3\), then, \(\{W_1^{(n)}\}_{n=1}^{\infty}\) goes to \(A_1\).
• 3. Let \(x_1 \in A_4\), then, \(\{W_1^{(n)}\}_{n=1}^{\infty}\) goes to \(A_1\).

Proof.

1. Let \(x_1^{(0)} \in A_1\). Then, \(0 \le x^{(0)}, y^{(0)}, z^{(0)} \le \frac{1}{2}\). Evidently, \(-1 \le 3x^{(0)} - 1 \le \frac{1}{2}\) can be verified. By squaring and adding \(-3(y^{(0)} - z^{(0)})^2\), the last inequality becomes \(0 \le (3x^{(0)} - 1)^2 - 3(y^{(0)} - z^{(0)})^2 \le 1\), and \(9(x^{(0)})^2 - 6x^{(0)} + 1 - 3(y^{(0)} - z^{(0)})^2 \le 1\) is obtained. By dividing the previous inequality by three after adding two to both parts of the inequality, we derive \(3(x^{(0)})^2 - 2x^{(0)} + 1 - (y^{(0)} - z^{(0)})^2 \le 1\).

Therefore, \[2(x^{(0)})^2 + (y^{(0)} + z^{(0)})^2 - (y^{(0)} - z^{(0)})^2 \le 1\].

Then, \(2(x^{(0)})^2 + 4y^{(0)}z^{(0)} \le 1\), which implies that \(x' \le \frac{1}{2}\). It is observable that

\(y' \le \frac{1}{2} \forall x_1^{(0)}\). Evidently, \(0 \le (y^{(0)})^2, (z^{(0)})^2 \le \frac{1}{4}\) is obtained, which implies that \(z' \le \frac{1}{2}\). Hence, \(A_1\) is an invariant region, i.e. if an initial point \(x_1^{(0)}\) is taken inside this region, then the trajectory will go to a point that is also in this region.

2. The second coordinate of \(W_1\) is less than \(\frac{1}{2}\) at any initial point \(x_1^{(0)}\), thereby indicating that \(A_3\) is not an invariant region. Subsequently, suppose that \(A_2\) is an invariant region, which indicates that \(y' \le x'\) and \(z' \le x'\). However,

\[x' = (x^{(0)})^2 + 2y^{(0)}z^{(0)} \le (x^{(0)})^2 + (y^{(0)})^2 + (z^{(0)})^2\]
\[\le x^{(0)}(x^{(0)} + y^{(0)} + z^{(0)}) = x^{(0)}.\]

Then \(\frac{x'}{x^{(0)}} < 1\), which implies that the first coordinate is a decreasing bounded sequence that converges to zero, thereby contradicting our assumption. Hence, if \(x_1^{(0)} \in A_2 \cup A_3\), then \(n_{k_1}, n_{k_2} \in \square\), such that the sequences \(x^{(n_{k_1})}\) and \(y^{(n_{k_2})}\) tend toward invariant region \(A_1\).

3. Suppose that \(A_4\) is an invariant region; hence, \(z' \ge y' + x'\) and \(x', y' \le \frac{1}{2}\). Evidently, \(x' \le y'\). By using the last inequality and the first coordinate of \(W_1\), we obtain \(y' \ge 2y^{(0)}z^{(0)}\). That is, \(z' \le \frac{1}{2}\), which repudiates our assumption. Hence, region \(A_4\) is not invariant.

Theorem 2. Let \(W_1: S^2 \to S^2\) be a \(\xi^{(as)}\)-QSO given by Eq. (10) and \(x_1^{(0)} = (x^{(0)}, y^{(0)}, z^{(0)}) \notin Fix(W_1)\) belongs to simplex \(S^2\) as initial point. In this case, the two statements below are valid:

1. One has \[Fix(W_1) = \left\{ e_1, e_3, \left( \frac{3 - \sqrt{3}}{4}, \frac{\sqrt{3}}{4}, \frac{1}{4} \right) \right\},\]

2. One has \[\omega_{W_1}(x_1^{(0)}) = \left\{ \left( \frac{3 - \sqrt{3}}{4}, \frac{\sqrt{3}}{4}, \frac{1}{4} \right) \right\}.\]

Proof. Let \(W_1: S^2 \to S^2\) be a \(\xi^{(as)}\)-QSO given by Eq. (10), \(x_1^{(0)} \notin Fix(W_1)\) be any beginning point that belongs to \(S^2\) and \(\{W_1^{(n)}\}_{n=1}^{\infty}\) be a trajectory of \(W_1\) that starts at \(\chi_1^{(0)}\).

1. The set of fixed points of \(W_1\) is obtained by finding the solution for the following system of equations:

\[= \begin{cases} x = x^2 + 2yz \\ y = 2x(1-x) \\ z = y^2 + z^2 \end{cases}\] (12)

Depending on the first equation in system Eq. (12), we derive \(x-x^2=2yz\). Subsequently, the last equation is multiplied by 2 and the new equation is substituted into the second equation in system Eq. (12). We obtain y(1-4z)=0 and find y=0 or \(z=\frac{1}{4}\). If y=0, then x=0 or x=1 can be easily found; hence, the fixed points are \(e_1=(1,0,0)\) and \(e_3=(0,0,1)\). If \(z=\frac{1}{4}\), then \(y=\frac{\sqrt{3}}{4}\) and \(x=\frac{3-\sqrt{3}}{4}\) can be found by using the first and third equation in system Eq. (12). Therefore, the fixed point is \(\left(\frac{3-\sqrt{3}}{4},\frac{\sqrt{3}}{4},\frac{1}{4}\right)\).

2. To investigate the behavior of \(\{W_1^{(n)}\}_{n=1}^{\infty}\), the following regions are introduced:

\[\begin{split} B_1 &\coloneqq \{x_1^{(0)} \in S^2 : 0 < z^{(0)} \le x^{(0)} < y^{(0)} < \frac{1}{2} \}, \\ B_2 &\coloneqq \{x_1^{(0)} \in S^2 : 0 \le y^{(0)} \le z^{(0)} \le x^{(0)} \le \frac{1}{2} \}, \\ B_3 &\coloneqq \{x_1^{(0)} \in S^2 : 0 < x^{(0)} \le y^{(0)} \le z^{(0)} \le \frac{1}{2} \}, \\ B_4 &\coloneqq \{x_1^{(0)} \in S^2 : 0 < z^{(0)} \le x^{(0)} \le \frac{1}{3}, \quad \frac{1}{3} < y^{(0)} \le \frac{1}{2} \}. \end{split}\]

Subsequently, the behavior of \(\left\{W_1^{(n)}\right\}_{n=1}^{\infty}\) across all the previously mentioned regions will be explored. After that, the behavior of \(\left\{W_1^{(n)}\right\}_{n=1}^{\infty}\) will be simple to describe.

Given that \(y', x' \leq \frac{1}{2}\), we can easily conclude \(x' \leq y'\), thereby indicating that \(B_2\) cannot possibly be an invariant region. Subsequently, we intend to verify whether \(B_3\) is or is not an invariant region. Let \(x_1^{(0)} \in B_3\). Then,

\[z' = (z^{(0)})^2 + (y^{(0)})^2 < (x^{(0)})^2 + (y^{(0)})^2 + (z^{(0)})^2 \le x^{(0)}z^{(0)} + y^{(0)}z^{(0)} + (z^{(0)})^2\]\[= z^{(0)}(x^{(0)} + y^{(0)} + z^{(0)}) = z^{(0)}\]

We determine that \(\frac{z'}{z^{(0)}} < 1\), which indicates that \(z^{(n)}\) is a decreasing bounded sequence. By this we mean that \(z^{(n)}\) converges to a fixed point of zero, thereby negating our presumption. Thus, region \(B_3\) is not invariant. Then, we consider a new sequence \(x' + z' = 2(x^{(0)})^2 - 2x^{(0)} + 1\). The new sequence has a minimum value of \(\frac{1}{2}\), which indicates that all coordinates are greater than zero and smaller than \(\frac{1}{2}\). Hence, if \(x_1^{(0)} \in B_2 \cup B_3 \cup A_1\), then \(n_{k_1}, n_{k_2}, n_{k_3} \in \square\), such that the sequences \(x^{(n_{k_1})}, y^{(n_{k_2})}\), and \(z^{(n_{k_3})}\) return to invariant region \(B_3\). Let \(x^{(0)} \leq \frac{1}{3}\). We can easily check whether the maximum value of the first coordinate \(x' = (x^{(0)})^2 + 2y^{(0)}(1 - x^{(0)} - y^{(0)})\) occurs when \(\left(\frac{1}{3}, \frac{1}{3}\right)\). In this case, \(x^{(n)} \leq \frac{1}{3}\) and \(z^{(n)} \leq \frac{1}{3}\). Given that all coordinates are equal to one, we conclude that \(y^{(n)} \geq \frac{1}{3}\). Therefore, if \(x_1^{(0)} \in B_1\), then \(n_k \in \mathbb{N}\), such that \(\left\{W_1^{(n_k)}\right\}_{n=1}^{\infty}\) returns to \(B_4\). Hence, \(B_4\) is an invariant region.

We have proven that if \(x_1^{(0)} \in B_i, i \in \{1,\dots,3\}\), then the trajectory \(\{W_1^{(n)}\}_{n=1}^\infty\) goes to invariant region \(B_4\). Thus, exploring the behavior of \(\{W_1^{(n)}\}_{n=1}^\infty\) over region \(B_4\) is adequate. Evidently, \(y^{(n)} \leq y^{(n+1)}\), i.e. it is a bounded increasing sequence. Given that \(y^{(n)} + x^{(n)}\) is a bounded decreasing sequence and \(x^{(n)} = y^{(n)} - y^{(n)} + x^{(n)}\), we conclude that \(x^{(n)}\) is a decreasing bounded sequence that converges to \(\frac{3-\sqrt{3}}{4}\). Thus, we have \(y^{(n)}\) converging to \(\frac{\sqrt{3}}{4}\). Therefore, \(\omega_{W_1}\left(x_1^{(0)}\right) = \{\left(\frac{3-\sqrt{3}}{4},\frac{\sqrt{3}}{4},\frac{1}{4}\right)\}\), which is the desired conclusion.

Let the following regions:

\[int(S^{2}) = \{x_{1}^{(0)} \in S^{2} : x^{(0)}y^{(0)}z^{(0)} > 0\},\]
\[int(S^{2}) = \{x_{1}^{(0)} \in S^{2} : x^{(0)}y^{(0)}z^{(0)} = 0\},\]

\[L_1 = \{x_1^{(0)} \in S^2 : x^{(0)} = 0\},\]
\[L_2 = \{x_1^{(0)} \in S^2 : y^{(0)} = 0\},\]
\[L_3 = \{x_1^{(0)} \in S^2 : z^{(0)} = 0\}\]

Theorem 3.

Let \(W_2: S^2 \to S^2\) be a \(\xi^{(as)}\)-QSO given by Eq. (11),

\(x_1^{(0)} = (x^{(0)}, y^{(0)}, z^{(0)}) \notin Fix(W_2) \cup Per_2(W_2)\) belongs to simplex \(S^2\) as initial point. Then, the three statements below are valid:

1. One has \(Fix(W_2) = \{e_3, (x^{\square}, y^{\square}, z^{\square})\}\), where

\[x^{\Box} = \frac{-1}{6} \sqrt[3]{t} - \frac{8}{3\sqrt[3]{t} + \frac{5}{3}},\] \[y^{\Box} = \frac{-1}{6} \frac{3\sqrt{17}\sqrt[3]{t} + 2\sqrt[3]{t^2} - 24\sqrt{17} - 5\sqrt[3]{t} - 88}{\sqrt[3]{t^2}},\] \[z^{\Box} = \frac{-1}{6} \frac{2\sqrt[3]{t^2} - 3\sqrt{17}\sqrt[3]{t} - 11\sqrt[3]{t}, +6\sqrt{17} - 10}{\sqrt[3]{t^2}}, \text{ and } t = (98 + 18\sqrt{17}).\]

2. One has \[Per_2(W_2) = \begin{cases} e_3, (0, y^\circ, 1 - y^\circ), & \text{if } x^{(0)} = 0 \\ e_3, (x^\circ, 0, 1 - x^\circ), & \text{if } y^{(0)} = 0 \end{cases}\]
where \(y^\circ = \frac{1}{6} (1 + 3\sqrt{57})^{\frac{1}{3}} - \frac{4}{3(1 + 3\sqrt{57})^{\frac{1}{3}}} + \frac{2}{3}\)
\(x^\circ = \frac{-1}{6} (46 + 6\sqrt{57})^{\frac{1}{3}} - \frac{2}{3(46 + 6\sqrt{57})^{\frac{1}{3}}} + \frac{4}{3}\)

3. One has

\[\omega_{w_{2}}(x_{1}^{(0)}) = \begin{cases} (x^{\square}, y^{\square}, z^{\square}) & ,if \quad x_{1}^{(0)} \in int(S^{2}) \\ (x^{\circ}, 0, 1 - x^{\circ}), (0, y^{\circ}, 1 - y^{\circ}) & ,if \quad x_{1}^{(0)} \in \overline{int(S^{2})} \\ e_{3} & ,if \quad x^{(0)}, y^{(0)} = 1 \end{cases}\]

Proof. Let \(W_2: S^2 \to S^2\) be a \(\xi^{(as)}\)-QSO given by Eq. (11), \(x_1^{(0)} \notin Fix(W_2) \cup Per_2(W_2)\) belongs to \(S^2\), and \(\{W_2^{(n)}\}_{n=1}^{\infty}\) be a trajectory of \(W_2\) beginning at \(x_1^{(0)}\).

1. The set of fixed points of \(W_2\) is obtained by finding the solution for the following system of equations:

\[\begin{cases} x = 2yz \\ y = 2x - x^2 \\ z = z^2 + y^2 \end{cases}\] (13)

Based on the first equation in system Eq. (13), we have \(z = \frac{x}{2y}\). By using z = 1 - y - x and the second equation in system Eq. (13), we obtain \(3x - 14x^2 + 10x^3 - 2x^4 3x - 14x^2 + 10x^3 - 2x^4 = 0\). Thus, the roots of the previous equation are \(\{0, x'\}\). By compensating for the values of x, namely, x = 0 and \(x = x^{\bullet}\) in the second equation in system Eq. (13), we obtain y = 0 and z = 1 or \(y = y^{\bullet}\) and \(z = z^{\bullet}\). Therefore, the fixed points of \(w^2\) are \(e_3\) and \((x^{\bullet}, y^{\bullet}, z^{\bullet})\).

2. To find the 2-periodic points of \(W_2\), we should prove that \(W_2\) has no specified order of periodic points in set \(S^2 \setminus L_1 \cup L_2\). Evidently, \(y^{(n+1)} \ge y^{(n)}\), i.e. the second coordinate of \(W_2\) increases along the iteration of \(W_2\) in set \(S^2 \setminus L_2\). Consider a new sequence \(x' + y' = 2x^{(0)} - (x^{(0)})^2 + 2y^{(0)}(1-x^{(0)}-y^{(0)})\). Whether x' + y' is a decreasing sequence can easily be checked, thereby indicating that sequence \(x^{(n)}\) is decreasing, because \(x^{(n)} = y^{(n)} + y^{(n)}\). Thus, the first coordinate of \(W_2\) decreases along the iteration of \(W_2\) in set \(S^2 \setminus L_1\), which indicates that \(W_2\) has no specified order 2-periodic points in set \(S^2 \setminus L_1 \cup L_2\). Therefore, finding the 2-periodic points of \(W_2\) in \(L_1 \cup L_2\) is sufficient. To find the 2-periodic points, the succeeding system of equations should be solved:

\[\begin{cases} x = 2(2x - x^2)(y^2 + z^2) \\ y = 4yz - 4y^2z^2 \\ z = (2x - x^2)(y^2 + z^2)^2 \end{cases}\] (14)

We start when x = 0. Then, we find the solution for \(y = 4y - 8y^2 + 8y^3 - 4y^4\). We obtain the following solution: y = 0 or \(y = y^\circ\). If y = 0, then z = 1. If \(y = y^\circ\), then \(z = 1 - y^\circ\). Therefore, \(e_3\) and \((0, y^\circ, 1 - y^\circ)\) are 2-periodic points. On the other hand, if y = 0, then the solutions for the equation x = 0.

\(2(2-x-x^2)(1-x)^2\) are x=0 or \(x=x^\circ\). If x=0, then z=0; if \(x=x^\circ\), then \(z=1-x^\circ\). Therefore, \(e_3\) and \((x^\circ,0,1-x^\circ)\) are 2-periodic points.

3. To investigate the behavior of \(W_2\), the following regions are introduced:

\[\begin{split} \ell_1 &:= \{x_1^{(0)} \in int\left(S^2\right) \colon 0 < x^{(0)}, \, y^{(0)}, z^{(0)} \leq \frac{1}{2} \}; \\ \ell_2 &:= \{x_1^{(0)} \in int\left(S^2\right) \colon 0 < x^{(0)} \leq z^{(0)} \leq y^{(0)} \leq \frac{1}{2} \}. \end{split}\]

Evidently, \(y' = 2x^{(0)} - (x^{(0)})^2 \ge x^{(0)}\) and \(x' = 2y^{(0)}z^{(0)} \le (y^{(0)})^2 + (z^{(0)})^2 = z'\), which indicates that \(x^{(n)} \le z^{(n)}\) and \(x^{(n)} \le y^{(n)}\). To prove that \(\ell_1\) is an invariant region, assume that \(y' \ge \frac{1}{2}\) by using the first coordinate of \(W_2\). Then we have \(x' = 2y^{(0)}z^{(0)}\), which implies that \(x' \ge z'\). This relation is a contradiction because \(x' \le z'\). Thus, \(y^{(n)} \le \frac{1}{2}\). Now, suppose that \(z' \ge \frac{1}{2}\). By using the first coordinate in \(W_2\), we obtain \(x' \ge y'\), which is also a contradiction. Therefore, \(\ell_1\) is an invariant region. Moreover, if \(x_1^{(0)} \in \overline{\ell_1}\) , then \(n_k \in \square\) , such that \(W_2^{(n_k)}\)returns to invariant region \(\ell_1\). Let us complete proving that \(\ell_2\) is an invariant assume that \(y' \le z'\), this we region. \(z' = (z^{(0)})^2 + (y^{(0)})^2 \le 2(z^{(0)})^2\), i.e. \(\frac{z'}{z^{(0)}} < 1\). Therefore, \(z^{(n)}\) is a decreasing bounded sequence. That is, \(z^{(n)}\) converges to the fixed point of zero. Moreover, \(y^{(n)}\) is an increasing bounded sequence. Thus, \(y^{(n)}\) converges to zero. Whether \(y^{(n)}\) converges to zero if \(x^{(n)}\) converges to zero can be checked. The result implies that the limiting point for \(W_2\) is empty, which is a contradiction. Thus, \(n_k \in \square\), such that \(z^{(n_k)}\) returns to invariant region \(z' \leq y'\), which proves that \(\ell_2\)is an invariant region. Moreover, if \(x_1^{(0)} \in \ell_1\), then \(n_k \in \mathbb{D}\), such that \(\{W_2^{(n_k)}\}_{n=1}^{\infty}\)returns to invariant region \(\ell_2\).

Accordingly, the behavior of \(\{W_2^{(n)}\}_{n=1}^{\infty}\) can be described. As discussed in the proof of the second part of this theorem, we determine that the first and second coordinates, namely, \(x^{(n)}\) and \(y^{(n)}\) are decreasing and increasing sequences, respectively. Thus, \(x^{(n)}\) and \(y^{(n)}\) converge to a certain fixed point. The first and second coordinates of \(W_2\) converge; thus, the third coordinate also converges. Between the two fixed points, the properties mentioned above of \(W_2\) are only

satisfied by point \((x^{\bullet}, y^{\bullet}, z^{\bullet})\). Therefore, the limiting point is \(\omega_{W_{\gamma}}(x_1^{(0)}) = (x^{\square}, y^{\square}, z^{\square}), \ \forall x_1^{(0)} \in int(S^2)\).

To explore the behavior of \(\{W_2^{(n)}\}_{n=1}^{\infty}\) when \(x_1^{(0)} \in \overline{int(S^2)}\), consider three cases, i.e. when \(x^{(0)} = 0\), \(y^{(0)} = 0\), and \(z^{(0)} = 0\). If \(x^{(0)} = 0\), then \(V^{(1)}((0, y^{(0)}, z^{(0)}))\)=(x',0,1-x') and \(V^{(2)}((0,y^{(0)},z^{(0)}))=(0,y',1-y')\). By applying this process to the next iteration, we determine that \(V^{(2n+1)}((0, y^{(0)}, z^{(0)})) = (x^{(2n+1)}, 0, 1 - x^{(2n+1)})\)and \(V^{(2n)}((0, y^{(0)}, z^{(0)})) = (0, y^{(2n)}, 1 - y^{(2n)})\). That is, the behavior of \(\{W_2^{(n)}\}_{n=1}^{\infty}\) in this case will be on the xz-plane if n is an odd iteration and on the yz-plane if nis an even iteration. When the preceding process is performed when \(y^{(0)} = 0\), \(V^{(2n+1)}((x^{(0)},0,z^{(0)})) = (0,y^{(2n+1)},1-y^{(2n+1)})\)\(V^{(2n)}((x^{(0)},0,z^{(0)})) = (x^{(2n)},0,1-x^{(2n)})\). That is, the behavior of \(\{W_2^{(n)}\}_{n=1}^{\infty}\) in this case will be on the yz-plane if n is an odd iteration and on the xz-plane if n is an even iteration. Through the same process, we determine that \(V^{(2n+1)}((x^{(0)},0,z^{(0)})) = (0,y^{(2n+1)},1-y^{(2n+1)})\) and \(V^{(2n)}((x^{(0)},0,z^{(0)})) = (x^{(2n)},0,z^{(0)})\)\(1-x^{(2n)}\)) when \(z^{(0)}=0\). This indicates that \(n_k\in \square\), such that the behavior of \(\{W_2^{(n)}\}_{n=1}^{\infty}\) in the case of \(z^{(0)}=0\) will be on the yz-plane if n is an odd iteration and on the xz-plane if n is an even iteration. Therefore, studying the two cases when \(x^{(0)} = 0\) and \(y^{(0)} = 0\) is sufficient. Starting with \(x^{(0)} = 0\), consider the following function:

\[y^{(2)} = \nu(y^{(0)}) = 4y^{(0)} - 8(y^{(0)})^2 + 8(y^{(0)})^3 - 4(y^{(0)})^4\] (15)

where \(y^{(0)} \in (0,1)\). Now, \(Fix(v) \cap (0,1) = \{y^{\circ}\}\) can be shown. Through simple calculations, \(v\left((0,\frac{1}{2}]\right) \subseteq [\frac{1}{2},1)\) can be found. Thus, we conclude that \([\frac{1}{2},1)\) is sufficient to study the dynamics of v at interval (0,1).

To investigate the behavior of v, the interval \([\frac{1}{2},1)\) is divided into three intervals as follows: \(I_1 = [\frac{1}{2},y^\circ]\) , \(I_2 = [y^\circ,\frac{1}{2}+\frac{1}{2}\sqrt{\sqrt{2}-1}]\), and \(I_3 = [\frac{1}{2}+\frac{1}{2}\sqrt{\sqrt{2}-1},1)\). Evidently, \(v(v(y^{(0)})) \geq y^{(0)}\) when \(y^{(0)} \in I_1\) and

\(\nu(\nu(y^{(0)})) \le y^{(0)}\) when \(y^{(0)} \in I_2\). Therefore, two cases should be discussed separately.

• a. For any \(n \in \square\), \(v^{(2n+2)}(y^{(0)}) \ge v^{(2n)}(y^{(0)}) \forall y^{(0)} \in I_1\) can be easily shown. Thus, \(v^{(2n)}(y^{(0)})\) is an increasing bounded sequence. Furthermore, \(v^{(2n)}(y^{(0)})\) converges to a fixed point of \(v^{(2)}\). \(y^{\circ}\) is also a fixed point of \(v^{(2)}\) and is the only possible point of the convergence trajectory. Hence, sequence \(y^{(2n)}\) converges to \(y^{\circ}\).
• b. Similarly, \(\left\langle v^{(2n+2)}(y^{(0)}) \leq v^{(2n)}(y^{(0)}) \right\rangle \forall y^{(0)} \in I_2\). Thus, \(v^{(2n)}\) is a decreasing bounded sequence. Furthermore, \(v^{(2n)}(y^{(0)})\) converges to a fixed point of \(v^{(2)}\). \(y^{\circ}\) is also a fixed point of \(v^{(2)}\) and it is the only possible point of the convergence trajectory. Hence, sequence \(y^{(2n)}\) converges to \(y^{\circ}\).

To explore the behavior of \(\nu\) when \(y^{(0)} \in I_3\), the following claim is required:

Claim: Let \[y^{(0)} \in I_3\]. Then, \(n_{\nu} \in \square\), such that \(v^{(n_k)} \in I_1 \cup I_2\).

Proof. Let \(y^{(0)} \in I_3\). Suppose that the interval \(I_3\) is an invariant interval, which indicates that \(y^{(n)} \in I_3\) for any \(n \in \square\). Evidently, \(v^{(n+1)}(y^{(0)}) \le v^{(n)}(y^{(0)})\), which results in \(v^{(n)}\) being a decreasing bounded sequence and converging to a fixed point v. However, \(Fix(v) \cap I_3 = \emptyset\), which is a contradiction. Hence, \(n_k \in \square\), such that \(v^{(n_k)} \in I_1 \cup I_2\).

In accordance with this claim, \(y^{(2n)}\) will go to \(I_1 \cup I_2\) after several iterations. Thus, sequence \((0, y^{(2n)}, z^{(2n)})\) converges to \((0, y^{\circ}, 1 - y^{\circ})\) whenever \(x^{(0)} = 0\).

Let \(y^{(0)} = 0\) and consider the following function:

\[x^{(2)} = \mathcal{G}(x^{(0)}) = 4x^{(0)} - 10(x^{(0)})^2 + 8(x^{(0)})^3 - 2(x^{(0)})^4, \tag{16}\] where \(x^{(0)} \in (0,1)\). When we do this, \(Fix(\vartheta) \cap (0,1) = \{x^{\circ}\}\) can be easily shown. Through simple calculations, we determine \(\vartheta\left(\left[0,1-\frac{1}{2}\sqrt{2}\right]\subseteq \left[1-\frac{1}{2}\sqrt{2},1\right)\right)\) and conclude that \(\left[1-\frac{1}{2}\sqrt{2},1\right)\) is sufficient to study the dynamics of \(\vartheta\) on (0,1).

To investigate the behavior of \(\vartheta\), the invariant interval \([1 - \frac{1}{2}\sqrt{2}, 1)\) is divided into three intervals as follows: \(I_1 = [1 - \frac{1}{2}\sqrt{2}, x^\circ]\), \(I_2 = [x^\circ, \frac{1}{2}]\), and \(I_3 = [\frac{1}{2}, 1)\).

Thus, we have two separate cases:

• a. Let \(x^{(0)} \in I_1\), then \(\mathcal{G}(x^{(0)}) \in I_2\) and, \(\mathcal{G}^{(2)}(x^{(0)}) \in I_1\). \(\mathcal{G}^{(2n+2)}(x^{(0)}) \leq \mathcal{G}^{(2n)}(x^{(0)})\) whenever \(x^{(0)} \in I_1\) can be easily checked. Therefore, \(\mathcal{G}^{(2n)}\) is a decreasing bounded sequence that converges to a fixed point of \(\mathcal{G}^{(2)}\). \(x^\circ\) is a fixed point of \(\mathcal{G}^{(2)}\) and the only possible point of the convergence trajectory. Hence, \(\mathcal{G}^{(2)}\) converges to \(x^\circ\).
• b. Similarly, \(\det x^{(0)} \in I_2\), then \(\mathcal{G}(x^{(0)}) \in I_1\) and \(\mathcal{G}^{(2)}(x^{(0)}) \in I_2\). \(\mathcal{G}^{(2n+2)}(x^{(0)}) \geq \mathcal{G}^{(2n)}(x^{(0)}) \text{ whenever } x^{(0)} \in I_2 \text{ can be easily checked.}\) Therefore, \(\mathcal{G}^{(2n)}\) is an increasing bounded sequence that converges to a fixed point of \(\mathcal{G}^{(2)}\). \(x^\circ I\) s a fixed point of \(\mathcal{G}^{(2)}\) and the only possible point of the convergence trajectory. Hence, \(\mathcal{G}^{(2n)}\) converges to \(x^\circ\).

To explore the behavior of \(\mathcal G\), when \(x^{(0)} \in I_3\), the following claim is required: Claim: Let \(x^{(0)} \in I_3\). Then, \(n_k \in \square\), such that \(\mathcal G^{(n_k)} \in I_1 \cup I_2\).

Proof. Let \(x^{(0)} \in I_3\). Suppose that interval \(I_3\) is invariant, which indicates that \(x^{(n)} \in I_3\) for any \(n \in \square\). Evidently, \(\mathcal{G}^{(n+1)}(x^{(0)}) \leq \mathcal{G}^{(n)}(x^{(0)})\), which results in sequence \(\mathcal{G}^{(n)}\) being decreasingly bounded and converging to a fixed point of \(\mathcal{G}\). However, \(Fix(\mathcal{G}) \cap I_3 = \emptyset\), which is contradiction. Hence, \(n_k \in \square\), such that \(\mathcal{G}^{(n_k)} \in I_1 \cup I_2\).

In accordance with the claim, \(x^{(n)}\) will go to \(I_1 \cup I_2\) after several iterations. Thus, sequence \((x^{(2n)},0,z^{(2n)})\) converges to \((x^{\circ},0,1-x^{\circ})\) whenever \(y^{(0)}=0\). Alternatively, if \(x^{(0)}=0\), then

\[V^{(n)}(W_2) = \begin{cases} (0, y^{\circ}, 1 - y^{\circ}) & \text{if } n = 2k \\ (x^{\circ}, 0, 1 - x^{\circ}) & \text{if } n = 2k + 1 \end{cases}\] (17)

If \(y^{(0)} = 0\), then

\[V^{(n)}(W_2) = \begin{cases} (0, y^{\circ}, 1 - y^{\circ}) & \text{,if } n = 2k + 1\\ (x^{\circ}, 0, 1 - x^{\circ}) & \text{,if } n = 2k \end{cases}\] (18)

From the preceding demonstrations, we observe that if \(x^{(0)} = 0\) and n is even, then the behavior of \(\{W_2^{(n)}\}_{n=1}^{\infty}\) occurs in \((0, y^{\circ}, 1-y^{\circ})\), which is equal to the behavior of \(\{W_2^{(n)}\}_{n=1}^{\infty}\) when \(y^{(0)} = 0\) and n is an odd iteration. If \(y^{(0)} = 0\) and n is an even iteration, then the behavior of \(\{W_2^{(n)}\}_{n=1}^{\infty}\) occurs in \((x^{\circ}, 0, 1-x^{\circ})\), which is equal to the behavior of \(W_2\) when \(x^{(0)} = 0\) and u is an odd iteration. Therefore, the limiting point of \(W_2\) consists of \((x^{\circ}, 0, 1-x^{\circ})\) and \((0, y^{\circ}, 1-y^{\circ})\) whenever \(x^{(0)} \notin int(S^2)\). If \(x^{(0)} = 1\), then the behavior of \(\{W_2^{(n)}\}_{n=1}^{\infty}\) reaches fixed point \(e_3\) after three iterations; if \(y^{(0)} = 1\), then the behavior of \(\{W_2^{(n)}\}_{n=1}^{\infty}\) reaches fixed point \(e_3\) after one iteration. Therefore, the limiting point in this case includes \(e_3\), which is the desired conclusion.

Subsequently, the behavior of operator \(\{V_{15}^{(n)}\}_{n=1}^{\infty}\) selected from class \(G_9\) is explored:

\[V_{15} := \begin{cases} x' = (1 - \alpha)(x^{(0)})^2 + 2y^{(0)}z^{(0)} \\ y' = \alpha(x^{(0)})^2 + 2x^{(0)}(1 - x^{(0)}) \\ z' = (z^{(0)})^2 + (y^{(0)})^2 \end{cases}\](19)

The operator \(V_{15}\) can be redrafted as a convex combination \(V_{15} = (1-\alpha)W_1 + \alpha W_2\), where \(W_1\) and \(W_2\) are equal to the operators given by Eq. (10) and Eq. (11), respectively.

Corollary 1. Let \(W_1\) be a \(\xi^{(as)}\)-QSO given by Eq. (10) and let \(x_1^{(0)} = (x^{(0)}, y^{(0)}, z^{(0)}) \notin Fix(W_1)\) belong to simplex \(S^2\) as initial point. Then, the two statements below are valid:

1. One has \[Fix(W_1) = \left\{ e_1, e_3, \left( \frac{3 - \sqrt{3}}{4}, \frac{\sqrt{3}}{4}, \frac{1}{4} \right) \right\}\]

2. One has \[\omega_{w_1}(x_1^{(0)}) = \left\{ \left( \frac{3 - \sqrt{3}}{4}, \frac{\sqrt{3}}{4}, \frac{1}{4} \right) \right\}.\]

Let \(W_2: S^2 \to S^2\) be a \(\xi^{(as)}\)-QSO given by Eq. (11) and let \(x_1^{(0)} = (x^{(0)}, y^{(0)}, z^{(0)}) \notin Fix(W_2)\) belong to simplex \(S^2\) as initial point. Then, the following statements hold true:

1. One has \(Fix(W_2) = \{e_3, (x^{\square}, y^{\square}, z^{\square})\}\).

where, \[x^{\Box} = \frac{-1}{6} \sqrt[3]{t} - \frac{8}{3\sqrt[3]{t} + \frac{5}{3}}\],

\[y^{\Box} = \frac{-1}{6} \frac{3\sqrt{17}\sqrt[3]{t} + 2\sqrt[3]{t^2} - 24\sqrt{17} - 5\sqrt[3]{t} - 88}{\sqrt[3]{t^2}},\] \[z^{\Box} = \frac{-1}{6} \frac{2\sqrt[3]{t^2} - 3\sqrt{17}\sqrt[3]{t} - 11\sqrt[3]{t} + 6\sqrt{17} - 10}{\sqrt[3]{t^2}}\] and \(t = (98 + 18\sqrt{17}).\)

2. One has \[Per_2(W_2)\] \(\left\{ e_3, (0, y^{\circ}, 1 - y^{\circ}), if \quad x^{(0)} = 0 \\ e_3, (x^{\circ}, 0, 1 - x^{\circ}), if \quad y^{(0)} = 0 \right\}\)

where, \[y^{\circ} = \frac{1}{6} (1 + 3\sqrt{57})^{\frac{1}{3}} - \frac{4}{3(1 + 3\sqrt{57})^{\frac{1}{3}}} + \frac{2}{3}\],

\[x^{\circ} = \frac{-1}{6} (46 + 6\sqrt{57})^{\frac{1}{3}} - \frac{2}{3(46 + 6\sqrt{57})^{\frac{1}{3}}} + \frac{4}{3}.\]

3. One has

\[\omega_{w_{2}}(x_{1}^{(0)}) = \begin{cases} (x^{0}, y^{0}, z^{0}) & ,if \quad x_{1}^{(0)} \in int(S^{2}) \\ (x^{\circ}, 0, 1 - x^{\circ}), (0, y^{\circ}, 1 - y^{\circ}), if \quad x_{1}^{(0)} \notin int(S^{2}) \\ e_{3} & ,if \quad x^{(0)}, y^{(0)} = 1 \end{cases}\]

5 Dynamics of classes \(G_{13}\) and \(G_{14}\)

In this section, we study the dynamics of \(V_{26,25}\): \(S^2 \to S^2\) selected from \(G_{14}\) and \(G_{13}\). To start, \(V_{26}\) is rewritten as follows:

\[V_{26} := \begin{cases} x' = (y^{(0)})^2 + (z^{(0)})^2 + 2x^{(0)} \left(1 - x^{(0)}\right) \\ y' = \alpha (x^{(0)})^2 + 2y^{(0)} z^{(0)} \\ z' = \left(1 - \alpha\right) (x^{(0)})^2 \end{cases}\](20)

The operator \(V_{26}\) can be redrafted as a convex combination \(V_{26} = \alpha W_1 + (1 - \alpha)W_2\), where

\[W_{1} := \begin{cases} x' = (y^{(0)})^{2} + (z^{(0)})^{2} + 2x^{(0)} (1 - x^{(0)}) \\ y' = (x^{(0)})^{2} + 2y^{(0)} z^{(0)} \\ z' = 0 \end{cases}\](21)

and

\[W_{2} := \begin{cases} x' = (y^{(0)})^{2} + (z^{(0)})^{2} + 2x^{(0)} (1 - x^{(0)}) \\ y' = 2y^{(0)}z^{(0)} \\ z' = (x^{(0)})^{2} \end{cases}\](22)

Theorem 4. Let \(W_1\) be a \(\xi^{(as)}\)-QSO given by Eq. (20) and \(x_1^{(0)} = (x^{(0)}, y^{(0)}, z^{(0)}) \notin Fix(W_1) \cup Per_2(W_1)\) belongs to simplex \(S^2\) as initial point. Then, the three statements below are valid:

1. One has \[Fix(W_1) = \left\{ \left( \frac{\sqrt{5}}{2} - \frac{1}{2}, \frac{3}{2} - \frac{\sqrt{5}}{2}, 0 \right) \right\}\]

2. One has \[Per_2(W_1) = \left\{ e_1, e_2, \left(\frac{\sqrt{5}}{2} - \frac{1}{2}, \frac{(-1 + \sqrt{5})^2}{4}, 0\right) \right\},\]

3. \[\omega_{W_1}(x_1^{(0)}) = \{e_1, e_3\}.\]

Proof. Let \(W_1: S^2 \to S^2\) be a \(\xi^{(as)}\)-QSO given by Eq. (22), \(x_1^{(0)} \notin Fix(W_1) \cup Per_2(W_1)\) belongs to \(S^2\), and \(\{W_1^{(n)}\}_{n=1}^{\infty}\) be a trajectory of \(W_1\) beginning from point \(x_1^{(0)}\).

1. The set of fixed points of \(W_1\) is obtained by finding the solution to the set of equations below:

\[\begin{cases} x = y^{2} + z^{2} + 2x(1-x) \\ y = x^{2} + 2yz \\ z = 0 \end{cases}\] (23)

By substituting the second and third equations Eq. (22) to the first equation, then the first equation in system Eq. (22) becomes \(x^4 - 2x^2 + x\), then x = 0, x = 1, and \(x = \frac{\sqrt{5}}{2} - \frac{1}{2}\). \(\frac{\sqrt{5}}{2} - \frac{1}{2}\) is verified as the only solution that satisfies system Eq. (22). Hence, the fixed point is only \(\left(\frac{\sqrt{5}}{2} - \frac{1}{2}, \frac{3}{2} - \frac{\sqrt{5}}{2}, 0\right)\).

2. Let \(x_1^{(0)} = (1,0,0)\). Then, \(V^{(1)}(x^0, y^0, z^0) = (0,1,0)\) and \(V^{(2)}(x^0, y^0, z^0) = (1,0,0)\), which indicates the presence of 2-periodic points. To find all the points, the following system of equations should be solved:

\[\begin{cases} x = 2x^{2} - x^{4} \\ y = (1 - (1 - y)^{2})^{2} \\ z = 0 \end{cases}\] (24)

From the first equation in system Eq. (23), \(x \in \left\{0,1,\frac{\sqrt{5}}{2} - \frac{1}{2}\right\}\) then \(y \in \left\{1,0,\frac{\left(-1+\sqrt{5}\right)^2}{4}\right\}\). Therefore, \(Per_2(W_1) = \left\{e_1,e_2,\left(\frac{\sqrt{5}}{2} - \frac{1}{2},\frac{\left(-1+\sqrt{5}\right)^2}{4},0\right)\right\}\).

3. Let \(x_1^{(0)} \notin Fix(W_1) \cup Per_2(W_1)\). \(L_3\) is an invariant line under \(W_1\). Thus, the behavior of \(\{W_1^{(n)}\}_{n=1}^{\infty}\) is explored over this line. Let \(x_1^{(0)} \in L_3\). Then, \(W_1\) becomes:

\[\begin{cases} x' = (y^{(0)})^2 + 2x^{(0)} (1 - x^{(0)}) \\ y' = (x^{(0)})^2 \\ z' = 0 \end{cases}\] (25)

In this case, the first coordinate of \(W_1\) exhibits the form \(x' = \varphi(x^{(0)}) = (1 - x^{(0)})^2 + 2x^{(0)}(1 - x^{(0)})\). Clearly, the function \(\varphi\) decreases on [0,1] and function \(\varphi^{(2)}\) is increasing on [0,1]. From the previous two steps, \(Fix(\varphi) \cap [0,1] = \left\{\frac{\sqrt{5}}{2} - \frac{1}{2}\right\}\) and \(ix(\varphi^2) \cap [0,1] = \left\{0, \frac{\sqrt{5}}{2} - \frac{1}{2}, 1\right\}\), which indicates that intervals \(\left[0, \frac{\sqrt{5}}{2} - \frac{1}{2}\right]\) and \(\left[\frac{\sqrt{5}}{2} - \frac{1}{2}, 1\right]\) are invariant under the function \((\varphi^2)\). Evidently, \(\varphi^{(2)}(x^{(0)}) \leq x^{(0)}\) for any \(x^{(0)} \in \left[0, \frac{\sqrt{5}}{2} - \frac{1}{2}\right]\) and \(\varphi^{(2)}(x^{(0)}) \geq x^{(0)}\) for any \(x^{(0)} \in \left[\frac{\sqrt{5}}{2} - \frac{1}{2}, 1\right]\). If \(x^{(0)} \in \left[0, \frac{\sqrt{5}}{2} - \frac{1}{2}\right]\), then \(\omega_{\varphi^{(2)}} = \{0\}\); if \(x^{(0)} \in \left[\frac{\sqrt{5}}{2} - \frac{1}{2}, 1\right]\), then \(\omega_{\varphi^{(2)}} = \{1\}\). In another way,

\[V^{(n)}(W_1) = \begin{cases} \left(\varphi^{(2k)}(x^{(0)}), 1 - \varphi^{(2k)}(x^{(0)}), 0\right) & \text{if } n = 2k \\ \left(\varphi^{(2k)}(\varphi(x^{(0)})), 1 - \varphi^{(2k)}(\varphi(x^{(0)})), 0\right) & \text{if } n = 2k + 1 \end{cases}\] (26)

Therefore, the limiting point is \(\omega_{w_1}\left(x_1^{(0)}\right) = \{e_1, e_2\}.\)

Theorem 5. Let \(W_2\) be a \(\xi^{(as)}\)-QSO given by Eq. (21) and \(x_1^{(0)} = (x^{(0)}, y^{(0)}, z^{(0)}) \notin Fix(W_2) \cup Per_2(W_2)\) belong to simplex \(S^2\) as initial point. Then, the two statements below are valid:

1. One has \(Fix(W_2) = \emptyset\). Moreover,

\[Per_2(W_2) = \{e_1, e_3, \left(\frac{\sqrt{5}-1}{2}, 0, \frac{1}{16}(\sqrt{5}-3)^2(\sqrt{5}+1)^2\right)\}.\]

2. One has \(\omega_{W_2}(x_1^{(0)}) = \{e_1, e_3\}\)

Proof. Let \(W_2: S^2 \to S^2\) be a \(\xi^{(as)}\)-QSO given by Eq. (21), \(x_1^{(0)} \notin Fix(W_2) \cup Per_2(W_2)\) belong to \(S^2\), and \(\left\{W_2^{(n)}\right\}_{n=1}^{\infty} = \frac{n!}{r!(n-r)!}\) be a trajectory of \(W_2\) starting at \(x_1^{(0)}\).

1. The set of fixed points of \(W_2\) is obtained by finding the solution for the equation set below:

\[\begin{cases} x = y^{2} + z^{2} + 2x(1-x) \\ y = 2yz \\ z = x^{2} \end{cases}\] (27)

The system provided by Eq. (26) has no solution on [0,1]. Therefore, the set of fixed points is \(\emptyset\). The second coordinate of \(W_2\) increases if \(z^{(n)} \ge \frac{1}{2}\)

and decreases if \(z^{(n)} \leq \frac{1}{2}\). In both cases, \(W_2\) has no specified order of periodic points in set \(W_2 \setminus L_2\), because the second coordinate of \(W_2\) increases or decreases along the iteration of \(W_2 \setminus L_2\). Therefore, finding the 2-periodic points of \(W_2 \setminus L_2\) over \(L_2\) is sufficient. To find the 2-periodic points of \(W_2\), the following system of equations should be solved:

\[\begin{cases} x = x^4 + 2x^2(1 - x^2) \\ y = 0 \\ z = (1 - (1 - z)^2)^2 \end{cases}\] (28)

The solution for the first equation in system Eq. (27) is easy to find. Therefore, the periodic points of \(W_2\) are \(e_1\), \(e_3\), and \(\left(\frac{\sqrt{5}-1}{2}, 0, \frac{1}{16}(\sqrt{5}-3)^2(\sqrt{5}+1)^2\right)\).

2. Let \(x_1^{(0)} \notin Fix(W_2) \cup Per_2(W_2)\) and \(y^{(0)} = 0\). The first coordinate of \(W_2\) can be rewritten as \(x' = (1 - x^{(0)})^2 + 2x^{(0)}(1 - x^{(0)})\) because the second coordinate is invariant over \(L_2\). The first coordinate is equal to the first coordinate of \(W_1\), which was proven in the previous theorem. Hence, we derive

\[V^{(n)}(W_2) = \begin{cases} \left(\varphi^{(2k)}(x^{(0)}), 0, 1 - \varphi^{(2k)}(x^{(0)})\right) & \text{if } n = 2k \\ \left(\varphi^{(2k)}(\varphi(x^{(0)})), 0, 1 - \varphi^{(2k)}(\varphi(x^{(0)}))\right) & \text{if } n = 2k + 1 \end{cases}\] (29)

Therefore, we determine that \(\omega_{W_2}\left(x_1^{(0)}\right)=\{e_1,e_3\}\). Let \(y^{(0)}\notin L_2\) and \(x^{(n)}<\frac{1}{2}\), which indicates that \(z^{(n)}<\frac{1}{2}\) and yields \(y^{(n+1)}< y^{(n)}\). If \(x^{(n)}>\frac{1}{2}\), then the third coordinate \(z^{(n)}\) is also smaller than \(\frac{1}{2}\), which indicates that \(y^{(n+1)}< y^{(n)}\). In the two previous cases, we conclude that \(\frac{y^{(n+1)}}{y^{(n)}}<1\), thereby making \(y^{(n+1)}\) is a decreasing bounded sequence that converges to zero, which indicates that studying the dynamics of \(W_2\) over \(L_2\) was sufficient. Therefore, \(\omega_{W_2}(x_1^{(0)})=\{e_1,e_3\}\) for any initial point \(x_1^{(0)}\) in \(S_2\).

Subsequently, we explore the behavior of \(\left\{V_{25}^{(n)}\right\}_{n=1}^{\infty}\), which is selected from class \(G_9\).

\[V_{25} := \begin{cases} x' = (y^{(0)})^2 + (y^{(0)})^2 + 2x^{(0)} (1 - x^{(0)}) \\ y' = \alpha (x^{(0)})^2 \\ z' = (1 - \alpha)(x^{(0)})^2 + 2y^{(0)}z^{(0)} \end{cases}\](30)

We rewrite \(V_{25}\) as a convex combination \(V_{25} = \alpha W_1 + (1 - \alpha)W_2\), where

\[W_{1} := \begin{cases} x' = (y^{(0)})^{2} + (y^{(0)})^{2} + 2x^{(0)} (1 - x^{(0)}) \\ y' = (x^{(0)})^{2} \\ z' = 2y^{(0)}z^{(0)} \end{cases}\](31)

and

\[W_{2} := \begin{cases} x' = (y^{(0)})^{2} + (y^{(0)})^{2} + 2x^{(0)} (1 - x^{(0)}) \\ y' = 0 \\ z' = (x^{(0)})^{2} + 2y^{(0)}z^{(0)} \end{cases}\](32)

Corollary 2 Let \(W_1: S^2 \to S^2\) given by Eq. (30) be a \(\xi^{(as)}\)-QSO and \(x_1^{(0)} = (x^{(0)}, y^{(0)}, z^{(0)}) \notin Fix(W_2) \cup Per_2(W_2)\) belong to simplex \(S^2\) as initial point. Then, the statements below are valid:

1. One has \(Fix(W_1) = \emptyset\). Moreover,

\[Per_2(W_1) = \{e_1, e_2, \left(\frac{\sqrt{5}-1}{2}, \frac{1}{16}\left(\sqrt{5}-3\right)^2\left(\sqrt{5}+1\right)^2, 0\right)\}.\]

2. One has \(\omega_{W_2}(x_1^{(0)}) = \{e_1, e_2\}\)

Let \(W_2: S^2 \to S^2\) given by Eq. (31) be a \(\xi^{(as)}\)-QSO. Then, the three statements below are valid:

• 1. One has \(Fix(W_2) = \left\{ \left( \frac{\sqrt{5}}{2} \frac{1}{2}, 0, \frac{3}{2} \frac{\sqrt{5}}{2} \right) \right\}\)
• 2. One has \(Per_2(W_2) = \{e_1, e_3\}\)
• 3. One has \(\omega_{W_2}(x_1^{(0)}) = \{e_1, e_2\}\)