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

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

$$\frac{d\mathbf{x}}{dt} = p\_3(y - u\_1\mathbf{x})(y - u\_2\mathbf{x})(y - u\_3\mathbf{x}),\\\frac{dy}{dt} = c(y - q\_1\mathbf{x})^2\tag{6}$$

$$p\_3 > 0, \ c > 0, \ u\_1 < u\_2 < u\_3.\\q(\in \mathbb{R}) \neq u\_i, i = \overline{1,3}.$$

The solution process includes the follows steps. Let us break the Eq. (6) family into subfamilies numbered r = 1, 4:

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

1. u1, u2, u3, q,

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

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`

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,

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

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

As a result, the topological type of phase portrait of all systems belonging to this 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

1. Eq. 4ð Þ–systems belonging to the number 1 family have in the Poincare circle Ω, 13

This means that in total, all large families of Eq:ð Þ4 dynamic systems of the A class may have 45

4. Family numbers 4, 5, and 6 have 5 different types of phase portraits per number.

7. Systems with 2 different multipliers in both right parts, belonging

In this section, the full solution of our task for Eq. (3) systems of the B class is given:

divided into m additional subfamilies (so as to say subsubfamilies) of the next order.

obtained subfamilies, and separatrices of singular points ∀s ¼ 1, 5.

mutual arrangement in Ω.

76 Differential Equations - Theory and Current Research

one common type of phase portrait:

Poincare circle Ω is invariable.

of its realization.

to a B class

in the Ω circle is common for the chosen subsubfamily.

A conclusion for the Section 6 of our chapter is:

3. Family number 3 have 10 types.

different topological types of phase portraits.

2. Eq:ð Þ4 – systems of the family number 2 have 7 types.

different topological types of phase portraits in a Poincare circle.

$$\mathbf{2.} \quad u\_1, u\_2, q, u\_3.$$


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

1. We study alternately the families of systems, r = 1,2, following the common program of Eq. (1) systems study [5], i.e.:

1. We fix r ∈f1; 2g,then we break the chosen family into subfamilies numbered s [5, 6], s = 1, 9, and find the topodynamical types (TD-types) of singular points of these systems.

10. Conclusions

phase portrait.

The presented work is devoted to the original study.

effective methods of investigation [8–10].

\* and Alexey Andreev<sup>2</sup>

2 St. Petersburg State University, St. Petersburg, Russia

Order Dynamic Systems. New York: Wiley; 1973

system. Differential Equations. 1997;33(5):702-703

10.3103/S1063454107020021, EID: 2-s2.0–84859730890

S0012266117080018, EID: 2-s2.0–85029534241

1 Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

\*Address all correspondence to: irandr@inbox.ru

students and postgraduates.

Author details

Irina Andreeva1

References

The main task of the work was to depict and describe all the different, in the topological meaning, phase portraits in a Poincare circle, possible for the dynamical differential systems belonging to a broad family of Eq. (1) systems, and to its numerical subfamilies. The authors have constructed all such phase portraits in two ways—in a descriptive (table) and in a graphic form. Each table contains 5–6 rows. Every row describes one invariant cell of the phase portrait in detail—it describes its boundary, source, and sink of its phase flow. The table was the descriptive

Phase Portraits of Cubic Dynamic Systems in a Poincare Circle

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

79

The second objective of this work was to develop, outline, and successfully apply some new

This was a theoretical work, but due to aforementioned new methods, the chapter may be useful for applied studies of dynamic systems of the second order with polynomial right parts. The authors hope that this work may be interesting and useful for researchers and for both

[1] Andronov AA, Leontovich EA, Gordon II, Maier AG. Qualitative Theory of Second-

[2] Andreev AF, Andreeva IA. On limit and separatrix cycles of a certain quasiquadratic

[3] Andreev AF, Andreeva IA. Local study of a family of planar cubic systems. Vestnik St. Petersburg University: Ser.1. Mathematics, Mechanics, Astronomy. 2007;2:11-16. DOI:

[4] Andreev AF, Andreeva IA, Detchenya LV, Makovetskaya TV, Sadovskii AP. Nilpotent centers of cubic systems. Differential Equations. 2017;53(8):1003-1008. DOI: 10.1134/


Then, we conclude the following.

For families of Eq. (6) systems with numbers 1, 2, 3, and 4, there exist

$$15 + 11 + 11 + 15 = 52$$

different topological types of phase portraits in a Poincare circle Ω.
