5. Systems containing 3 and 2 multipliers in their right parts

In this section, we present a solution to the main assigned problem for those Eq. (1) systems whose decompositions of forms X (x, y), Y (x, y) into real forms of lower degrees contain 3 and 2 multipliers, respectively:

$$\mathbf{X}(\mathbf{x},y) = p\_3(y - u\_1\mathbf{x})(y - u\_2\mathbf{x})(y - u\_3\mathbf{x}),\\\mathbf{Y}(\mathbf{x},y) = \mathbf{c}\left(y - q\_1\mathbf{x}\right)(y - q\_2\mathbf{x})\tag{2}$$

where p<sup>3</sup> > 0, c > 0, u<sup>1</sup> < u<sup>2</sup> < u3, q<sup>1</sup> < q2, ui 6¼ qj for each i and j.

The solution process contains the follows steps.

### 5.1. Basic concepts and notation

The following notations are introduced for the arbitrary system under consideration in the Section 5.

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

$$P(\mathbf{u}) \colon= X(1,\boldsymbol{\mu}) \equiv p\_3(\boldsymbol{\mu} - \boldsymbol{\mu}\_1)(\boldsymbol{\mu} - \boldsymbol{\mu}\_2)(\boldsymbol{\mu} - \boldsymbol{\mu}\_3), \ Q(\boldsymbol{\mu}) \colon= Y(1,\boldsymbol{\mu}) \equiv c(\boldsymbol{\mu} - q\_1)(\boldsymbol{\mu} - q\_2)$$

RSP (RSQ) – an ascending sequence of all real roots of then system's polynomial P(u) (Q(u)), RSPQ – an ascending sequence of all real roots of both the system's polynomials P(u), Q(u).

### 5.2. The double change (DC) transformation

Let us call a double change of variables in this dynamic system: (t, y) ! (�t, �y). The double change transformation transforms the system under consideration into another such system, for which numberings and signs of roots of polynomials P(u), Q(u), as well as the direction of motion upon trajectories with the increasing of t are reversed. Let us agree to call a pair of different Eq. (2) systems mutually inversed in relation to the DC transformation, if this transformation appears to convert one into another, and call them independent of a DC transformation in the opposite case.

Clearly, 10 different types of RSPQ are possible for an arbitrary Eq. (2) system, as C<sup>2</sup> <sup>5</sup> <sup>¼</sup> <sup>5</sup>! <sup>3</sup>!2! = 10. 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 inversed one among the first six Eq. (2)-systems.

6. Two classes of systems containing various combinations of two

ð Þ <sup>y</sup> � <sup>u</sup>2<sup>x</sup> <sup>k</sup><sup>2</sup>

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

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 ,

It is natural to distinguish two classes of Eq. (3) systems. The A class contains systems with

In this section, we give a full solution of the assigned task for systems belonging to the A class

2 , dy

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

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

An r-family of Eq:ð Þ4 – systems is the totality of Eq. 4ð Þ dynamic systems with the RSPQ

The steps of research of every fixed family belonging to Eq. 4ð Þ dynamic systems are as follows. 1. For all singular points of a given dynamic system that belongs to the family under consideration, let us introduce notions of S (saddle) and N (node) bundles of semitrajectories, which are adjacent to a chosen singular point; also let us introduce a notion

, Q uð Þ∶ ¼ Yð Þ� 1; u q u � q<sup>1</sup>

,Yð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>q</sup> <sup>y</sup> � <sup>q</sup>1<sup>x</sup> <sup>y</sup> � <sup>q</sup>2<sup>x</sup> (3)

Phase Portraits of Cubic Dynamic Systems in a Poincare Circle

http://dx.doi.org/10.5772/intechopen.75527

75

dt <sup>¼</sup> <sup>q</sup> <sup>y</sup> � <sup>q</sup>1<sup>x</sup> <sup>y</sup> � <sup>q</sup>2<sup>x</sup> (4)

<sup>u</sup> � <sup>q</sup><sup>2</sup>

<sup>4</sup> <sup>¼</sup> <sup>4</sup>!

,

<sup>2</sup>!2! = 6. Let us number

different multipliers in both right parts: an A-class

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:

dt <sup>¼</sup> p yð Þ � <sup>u</sup>1<sup>x</sup> ð Þ <sup>y</sup> � <sup>u</sup>2<sup>x</sup>

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

P(u) and Q(u). There exist 6 different possible variants of RSPQ as C<sup>2</sup>

A consistent research of families of Eq. 4ð Þ dynamic systems.

for its separatrix and a notion for its topodynamical type (TD-type).

<sup>X</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> p yð Þ � <sup>u</sup>1<sup>x</sup> <sup>k</sup><sup>1</sup>

dx

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

Now let us put into use an important notion:

number r from the list of six allowable variants.

them from 1 to 6 in some order.

P uð Þ∶ ¼ Xð Þ� 1; u p uð Þ � u<sup>1</sup> ð Þ u � u<sup>2</sup>

contain two multipliers each:

k1, k<sup>2</sup> ∈ N, k<sup>1</sup> þ k<sup>2</sup> =3.

of the Eq. (3) family, i.e.,

this chapter.

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 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.

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

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 DCtransformation of families, r = 1, 4.

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


From this section, we can conclude the following:

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

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

Systems of families 4 and 5: there exist 7 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 family includes an uncountable number of differential systems.
