6. Two classes of systems containing various combinations of two different multipliers in both right parts: an A-class

In Sections 6 and 7, the problem has been solved for an Eq. (3) family. The Eq. (3) family of Eq. (1) systems is as follows—the family consists of a totality of all Eq. (1) systems; for each of them, decompositions of forms X (x, y), Y (x, y) into real multipliers of the lowest degrees contain two multipliers each:

$$\mathbf{X}(\mathbf{x},y) = p(y - \mu\_1 \mathbf{x})^{\mathbf{k}} \left( y - \mu\_2 \mathbf{x} \right)^{\mathbf{k}}, \\ \mathbf{Y}(\mathbf{x},y) = \mathbf{q}(y - q\_1 \mathbf{x}) \left( y - q\_2 \mathbf{x} \right) \tag{3}$$

where p, q, u1, u2, q1, q<sup>2</sup> ∈ R, p > 0, q > 0, u<sup>1</sup> < u2, q<sup>1</sup> < q2, ui 6¼ qj for each i,j ∈ f g 1; 2 , k1, k<sup>2</sup> ∈ N, k<sup>1</sup> þ k<sup>2</sup> =3.

It is natural to distinguish two classes of Eq. (3) systems. The A class contains systems with k<sup>1</sup> ¼ 1, k<sup>2</sup> ¼ 2; and the B class contains systems with k<sup>1</sup> ¼ 2, k<sup>2</sup> ¼ 1:

In this section, we give a full solution of the assigned task for systems belonging to the A class of the Eq. (3) family, i.e.,

$$\frac{d\mathbf{x}}{dt} = p(y - u\_1\mathbf{x})(y - u\_2\mathbf{x})^2,\\\frac{dy}{dt} = \mathbf{q}(y - q\_1\mathbf{x})\left(y - q\_2\mathbf{x}\right) \tag{4}$$

The process of forming the solution contains steps similar to the ones described in Section 4 of this chapter.

For an arbitrary Eq:ð Þ4 – system, we introduce the following concepts.

Let P(u), Q(u) be the system's polynomials P, Q:

As we can conclude using the DC transformation of Eq. (2) systems, six of the RSPQs appear to be independent in pairs. Similarly, each of the remaining four systems has the mutually

Let us assign a specific number r ∈f g 1;…; 10 to each one of the different RSPQs of the Eq. (2) system in such a manner that RSPQr = 1, 6 are independent in pairs, while RSPQ sequences

An r family of Eq. (2) systems ∶ ¼ the totality of systems (belonging to Eq. (2) family) having

Now following a single plan, we consistently investigate the families of Eq. (2)systems that have numbers r = 1, 6: For families having numbers r = 7, 10, we obtain data through the DC-

1. We determine a list of singular points of systems of the fixed family in a Poincare circle Ω:

2. Further, we split the family under consideration to subfamilies with numbers s = 1, 7: For every subfamily, we reveal topodynamical types of singular points and separatrices of them.

3. We investigate the separatrices' behavior for all singular points of systems belonging to the chosen subfamily ∀ s∈ {1, …, 7}. Very important are the following questions: a question of a uniqueness of a continuation of every given separatrix from a small neighborhood of a singular point to all the lengths of this separatrix, as well as a question about a mutual arrangement of all separatrices in a Poincare circle Ω. We answer these questions for all

4. As a result of all previous studies, we depict phase portraits of dynamic systems of a given

family and outline the criteria of every portrait appearance [5, 6].

Systems of the family number r = 1 have 25 different types of phase portraits.

Systems of families 4 and 5: there exist 7 types of phase portraits per family.

family includes an uncountable number of differential systems.

Systems of families number 2 and 3: there are 9 types of phase portraits per family.

Systems belonging to the family number r = 6 show 36 different types of phase portraits.

Hence, we have obtained 93 different types in total for the systems described in this section—a lot of possible types at first glance. However, it is important to keep this in mind: every given

point in the list, we use the notions of a saddle (S) and node (N) bundles adjacent to this point's semi-trajectories, of a separatrix of the singular point, and of a topodynamical type

<sup>i</sup> (ui, 0) ∈ Г,i= 0, 3, u0= 0. For every

with numbers r = 7, 10 are mutually inversed to RSPQ`s which have numbers r = 1, 4: It is time to introduce the important notion of a family number r of Eq. (2) systems.

A plan of the investigation of each selected Eq. (2) family contains the follows items.

They appear to be a point O (0, 0)∈ Ω and points O�

inversed one among the first six Eq. (2)-systems.

74 Differential Equations - Theory and Current Research

the RSPQ number r:

transformation of families, r = 1, 4.

of the singular point (TD type).

families of systems under consideration.

From this section, we can conclude the following:

$$\mathbf{P}(\mathbf{u}) \colon= X(1, u) \equiv p(u - u\_1)(u - u\_2)^2, \quad \mathbf{Q}(u) \colon= Y(1, u) \equiv q(u - q\_1 \ )(u - q\_2),$$

and RSP (RSQ) be an ascending sequence of all the real roots of the system's polynomial, while P(u) (Q(u)),RSPQ is an ascending sequence of all the real roots of both system's polynomials P(u) and Q(u). There exist 6 different possible variants of RSPQ as C<sup>2</sup> <sup>4</sup> <sup>¼</sup> <sup>4</sup>! <sup>2</sup>!2! = 6. Let us number them from 1 to 6 in some order.

Now let us put into use an important notion:

An r-family of Eq:ð Þ4 – systems is the totality of Eq. 4ð Þ dynamic systems with the RSPQ number r from the list of six allowable variants.
