10. Conclusions

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.

2. We construct for the systems of a fixed subfamily ∀s ¼ 1, 9, the so-called off-road map (ORM) [5–7]. The ORM helps us to find an α ωð Þ� limit set of every α ωð Þ� separatrix. It also lets us describe the mutual arrangement of all separatrices in the Poincare circle Ω.

3. We depict all possible topologically different phase portraits for Eq. (6) systems.

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

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

<sup>x</sup>\_ <sup>¼</sup> <sup>p</sup>0x<sup>3</sup> <sup>þ</sup> <sup>p</sup>1x<sup>2</sup>

where p<sup>3</sup> > 0, c > 0, u<sup>1</sup> < u2, q ð Þ ∈R 6¼ u1, 2.

P uð Þ∶ ¼ Xð Þ� 1; u p3ð Þ u � u<sup>1</sup>

and there exists 3 different variants for their RSPQs.

9. Systems containing 2 and 1 different multipliers in right parts

portraits for the families 3 and 4.

78 Differential Equations - Theory and Current Research

Then, we conclude the following.

systems of the kind

nomials P, Q:

2. We investigate consistently families of Eq. (6) systems, r = 3, 4, using the DC transformation of the results obtained for families, r = 2, 1. Then, we depict all types of existing phase

15 + 11 + 11 + 15 = 52

In this section, we give the full solution of the problem for Eq. (7) systems, i.e., for the Eq. (1)

<sup>y</sup>\_ <sup>¼</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> bxy <sup>þ</sup> cy<sup>2</sup> � c yð Þ � qx

The process of study of these systems is quite similar to that previously described for other families of Eq. (1) systems. For an arbitrary Eq. (7) system, P(u), Q(u) are the system's poly-

We`ve revealed, that for every possible family of Eq. (7) systems, 7 different topological types of their phase portraits are being implemented. This means that for all three existing families of

2

A conclusion from our research for this particular type of systems is the following.

such systems, r = 1, 3, the number of different phase portraits is 21 [8, 9].

<sup>y</sup> <sup>þ</sup> <sup>p</sup>2xy<sup>2</sup> <sup>þ</sup> <sup>p</sup>3y<sup>3</sup> � <sup>p</sup>3ð Þ <sup>y</sup> � <sup>u</sup>1<sup>x</sup>

2

2 ,

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

ð Þ y � u2x (7)

2 , The presented work is devoted to the original study.

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 portrait.

The second objective of this work was to develop, outline, and successfully apply some new effective methods of investigation [8–10].

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 students and postgraduates.
