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

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

2. Now the considered family must be divided into subfamilies numbered s∈f g 1;…; 5 :Then it is necessary to determine the TD-types of singular points of systems belonging to the obtained subfamilies, and separatrices of singular points ∀s ¼ 1, 5.

dx

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

RSPQ shows 6 different variants, because C<sup>2</sup>

dx

portraits.

numbered r = 1, 4:

1. u1, u2, u3, q, 2. u1, u2, q, u3, 3. u1, q,u2, u3, 4. q,u1, u2, u3:

the list of possible RSPQs.

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

2

For an arbitrary Eq:ð Þ5 – system, P(u), Q(u) are the system's polynomials P, Q.

8. Systems containing 3 and 1 different multipliers in right parts

dt <sup>¼</sup> <sup>p</sup>3ð Þ <sup>y</sup> � <sup>u</sup>1<sup>x</sup> ð Þ <sup>y</sup> � <sup>u</sup>2<sup>x</sup> ð Þ <sup>y</sup> � <sup>u</sup>3<sup>x</sup> ,

2

ð Þ y � u2x ,

dy

<sup>4</sup> ¼ 6. We can thus conclude that all Eq. 5ð Þ family of systems is split into 52 different subfamilies, and all systems of each chosen subfamily show in a circle Ω, one common type of a phase portrait belonging to this particular subfamily. We have constructed all 52 topologically different phase

In this section, we solve the problem for an Eq. (6) family, i.e., for a family of Eq. (1) systems

p<sup>3</sup> > 0, c > 0, u<sup>1</sup> < u<sup>2</sup> < u3, qð Þ ∈ R 6¼ ui, i ¼ 1, 3:

The solution process includes the follows steps. Let us break the Eq. (6) family into subfamilies

Each of these is a totality of systems with an RSPQ number r, where r is the system's number in

Applying to the Eq. (6) system, a double change of variables (DC): (t, y) ! (�t,-y), we reveal that it transforms families of these systems having the numbers r = 1, 2, 3, 4, into their families with numbers r = 4, 3, 2, 1 respectively, and backward. We emphasize: this fact means that families of Eq. (6) systems having numbers 1 and 2 are not connected with the DC transformation, and that families having numbers 3 and 4 are not related to each other; at the same time, family number 3 is mutually inversed by the DC transformation to the family number 2, and family number 4 is mutually inversed to the family number 1 correspondingly. This conclusion

1. We study alternately the families of systems, r = 1,2, following the common program of

follows from the consideration of their RSPQ sequences [5, 6].

Eq. (1) systems study [5], i.e.:

ð Þ u � u<sup>2</sup> , Q uð Þ∶ ¼ Yð Þ� 1; u q u � q<sup>1</sup>

dy

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> : (5)

Phase Portraits of Cubic Dynamic Systems in a Poincare Circle

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

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

dt <sup>¼</sup> c y � <sup>q</sup>1<sup>x</sup> <sup>2</sup> (6)

,

77

3. For all five subfamilies, we investigate the separatrices` of singular points behavior and find an answer to a question concerning a uniqueness of a global continuation of every chosen separatrix from a tiny neighborhood of a singular point to all the lengths of this separatrix in the Poincare circle Ω, as well as an answer to a question of all separatrices` mutual arrangement in Ω.

The mutual arrangement of all separatrices in the Poincare circle is invariable when, for a given s, a global continuation of every separatrix of each singular point of the subfamily number s is unique. Consequently, all systems of a chosen subfamily number s have, in a Poincare circle, one common type of phase portrait:

But in a different situation, when, for a fixed number s, systems of such subfamily have, for example, m separatrices with global continuations that are not unique, this subfamily is divided into m additional subfamilies (so as to say subsubfamilies) of the next order.

As we could understand conducting their further study, for each of subsubfamilies, the global continuation of every separatrix is unique, and the mutual arrangement of separatrices in the Poincare circle Ω is invariable.

As a result, the topological type of phase portrait of all systems belonging to this subsubfamily in the Ω circle is common for the chosen subsubfamily.

4. We depict phase portraits in Ω for the systems of Eq:ð Þ4 families, r = 1, 6, in the two possible forms (the table and the graphic ones), and indicate for each portrait close to coefficient criteria of its realization.

A conclusion for the Section 6 of our chapter is:


This means that in total, all large families of Eq:ð Þ4 dynamic systems of the A class may have 45 different topological types of phase portraits in a Poincare circle.
