4. Infinitely remote singular points (IR points)

Now it is time to discuss the behavior of trajectories of the Eq. (1) systems in a neighborhood of infinity. For the investigation of this question we use the method of Poincare consecutive transformations, or mappings [1].

The first Poincare transformation

$$\mathbf{x} = \frac{1}{z}, \quad \mathbf{y} = \frac{u}{z} \quad \left( u = \frac{y}{x}, \quad z = \frac{1}{x} \right).$$

unambiguously maps a phase plane R<sup>2</sup> x,y of the Eq. (1) system onto a Poincare sphere ∑: <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>z</sup><sup>2</sup> <sup>¼</sup> 1 (where <sup>z</sup> ¼ �<sup>Z</sup> ½ � <sup>1</sup> ) with the diametrically opposite points identified, which is considered without its equator E, and an infinitely remote straight line of a plane R<sup>2</sup> x, <sup>y</sup> . The first Poincare transformation maps onto the equator E of the sphere ∑; the diametrically opposite points are also considered to be identified.

The Eq. (1) system in this mapping transforms into a system, which in the Poincare coordinates u, z after a time change dt ¼ �z<sup>2</sup>d<sup>τ</sup> looks like the following:

$$\frac{du}{d\tau} = P(u)u - Q(u)z, \qquad \frac{dz}{d\tau} = P(u)z,$$

where P uð Þ :� Xð Þ 1; u and Qð Þ u :� Yð Þ 1; u are reciprocal polynomials.

This new system is determined on the whole sphere ∑, including its equator, and on the whole ð Þ� <sup>u</sup>; <sup>z</sup> plane <sup>α</sup><sup>∗</sup>, which is tangent to a sphere <sup>∑</sup> at point C = (1, 0, 0). We shall study this system, namely on a plane R<sup>2</sup> u, <sup>z</sup> , and project the received results onto a closed circle Ω, sequentially mapping, first, a plane R<sup>2</sup> u,z onto the sphere ∑ from its center, and second, its lower semi-sphere H onto the Poincare circle Ω, i. e., onto a closed unit circle of a plane R<sup>2</sup> x,y through the orthogonal mapping.

For our new system, the axis z ¼ 0 is invariant (consists of this system's trajectories). On this axis, lie its singular points Oi ui ð Þ ; 0 , i ¼ 0, m, where ui, i ¼ 1, m are all real roots of the polynomial P uð Þ, and u<sup>0</sup> ¼ 0; at the same time, there may exist i<sup>0</sup> ∈ f1;…; mg: ui<sup>0</sup> = 0. Let us call such points IR points of the first kind for the Eq. (1) system.

The second Poincare transformation

$$x = \frac{v}{z}, \quad y = \frac{1}{z} \left(v = \frac{x}{y}, \ z = \frac{1}{y}\right)$$

also unambiguously maps a phase plane R<sup>2</sup> x,y onto a Poincare sphere ∑ with the diametrically opposite points identified, considered without its equator. Every Eq. (1) system transforms into a system, which in the coordinates τ, v, z looks like the following:

for every Eq. (1) system, the singular point O(0, 0) has not more than four separatrices (actually

Now it is time to discuss the behavior of trajectories of the Eq. (1) systems in a neighborhood of infinity. For the investigation of this question we use the method of Poincare consecutive

<sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>z</sup><sup>2</sup> <sup>¼</sup> 1 (where <sup>z</sup> ¼ �<sup>Z</sup> ½ � <sup>1</sup> ) with the diametrically opposite points identified, which is

Poincare transformation maps onto the equator E of the sphere ∑; the diametrically opposite

The Eq. (1) system in this mapping transforms into a system, which in the Poincare coordinates

This new system is determined on the whole sphere ∑, including its equator, and on the whole ð Þ� <sup>u</sup>; <sup>z</sup> plane <sup>α</sup><sup>∗</sup>, which is tangent to a sphere <sup>∑</sup> at point C = (1, 0, 0). We shall study this

lower semi-sphere H onto the Poincare circle Ω, i. e., onto a closed unit circle of a plane R<sup>2</sup>

For our new system, the axis z ¼ 0 is invariant (consists of this system's trajectories). On this axis, lie its singular points Oi ui ð Þ ; 0 , i ¼ 0, m, where ui, i ¼ 1, m are all real roots of the polynomial P uð Þ, and u<sup>0</sup> ¼ 0; at the same time, there may exist i<sup>0</sup> ∈ f1;…; mg: ui<sup>0</sup> = 0. Let us call

> <sup>v</sup> <sup>¼</sup> <sup>x</sup> y

; <sup>z</sup> <sup>¼</sup> <sup>1</sup> y

<sup>u</sup> <sup>¼</sup> <sup>y</sup> x , z <sup>¼</sup> <sup>1</sup> x

:

x,y of the Eq. (1) system onto a Poincare sphere ∑:

x, <sup>y</sup> . The first

x,y

dz

<sup>d</sup><sup>τ</sup> <sup>¼</sup> P uð Þz,

u, <sup>z</sup> , and project the received results onto a closed circle

u,z onto the sphere ∑ from its center, and second, its

4. Infinitely remote singular points (IR points)

<sup>x</sup> <sup>¼</sup> <sup>1</sup> z

u, z after a time change dt ¼ �z<sup>2</sup>d<sup>τ</sup> looks like the following: du

such points IR points of the first kind for the Eq. (1) system.

<sup>x</sup> <sup>¼</sup> <sup>v</sup> z , y <sup>¼</sup> <sup>1</sup> z

, y <sup>¼</sup> <sup>u</sup> z

considered without its equator E, and an infinitely remote straight line of a plane R<sup>2</sup>

<sup>d</sup><sup>τ</sup> <sup>¼</sup> P uð Þ<sup>u</sup> � Q uð Þz,

where P uð Þ :� Xð Þ 1; u and Qð Þ u :� Yð Þ 1; u are reciprocal polynomials.

2, 3, or 4 ones).

transformations, or mappings [1]. The first Poincare transformation

70 Differential Equations - Theory and Current Research

unambiguously maps a phase plane R<sup>2</sup>

points are also considered to be identified.

system, namely on a plane R<sup>2</sup>

through the orthogonal mapping.

The second Poincare transformation

Ω, sequentially mapping, first, a plane R<sup>2</sup>

$$\frac{dv}{d\tau} = -X(v,1) + Y(v,1)vz, \qquad \frac{dz}{d\tau} = Y(v,1)z^2.$$

This last system is determined on the whole sphere <sup>∑</sup>, and on the whole ð Þ� <sup>v</sup>; <sup>z</sup> plane <sup>α</sup>b, which is tangent to a spherePat point <sup>D</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>1</sup>; <sup>0</sup> [1]. A set <sup>z</sup> <sup>¼</sup> 0 is invariant for this last system. On this set, lie its singular pointsð Þ v0; 0 , where v<sup>0</sup> is any real root of the polynomial X vð Þ� ; <sup>1</sup> <sup>p</sup><sup>3</sup> <sup>þ</sup> <sup>p</sup>2<sup>v</sup> <sup>þ</sup> <sup>p</sup>1v<sup>2</sup> <sup>þ</sup> <sup>p</sup>0v<sup>3</sup>: It would be natural to call such points IR points of the second kind for Eq. (1) systems, but each of these points, for which v<sup>0</sup> 6¼ 0, obviously coincides with one of the IR-points of the first kind, namely with the point <sup>1</sup> <sup>v</sup><sup>0</sup> ; 0 � �,

while v<sup>0</sup> ¼ 0 is not a root of the polynomial X(x, 1), because X(0, 1) = p36¼ 0 for the Eq. (1) system. Consequently, the following corollary is correct.

Corollary 3. The infinitely remote singular points of any Eq. (1) system are only IR-points of the first kind.

With the orthogonal projection of a closed lower semi-sphere H of a Poincare sphere ∑ onto a plane x, y, its open part H one-to-one maps onto an open Poincare circle Ω, while its boundary E (an equator of the Poincare sphere ∑) maps onto the boundary of the Poincare circle Γ¼∂Ω, which implies the following. 1) Trajectories of any Eq.(– (including its singular point O (0, 0)) are displayed in a circle Ω, filling it.

2) Such a system's infinitely remote trajectories (including IR points) are displayed on the boundary Γ of a circle Ω, filling it.

Following Poincare, we call the first trajectories of the Eq. (1) system in Ω, and the second, we call trajectories of the Eq. (1) system on Γ.

As it follows from the aforementioned conclusions, to each IR point Oi ui ð Þ ; 0 , of the Eq. (1) system, i∈f1;…; mg, correspond two diametrically opposite points situated on the Γ circle.

$$\left. \mathcal{O}\_i^{\pm}(\mu\_i, 0) : \mathcal{O}\_i^+ \left( \mathcal{O}\_i^- \right) \in \Gamma^{+ (-)} := \left. \Gamma \right|\_{x > 0 \ (x < 0)} \right.$$

∀i ∈ 1;…; mg for the point O<sup>þ</sup> <sup>i</sup> O� i � �, � we shall introduce the following notation.


3. A notation of a word A<sup>þ</sup> <sup>i</sup> (A� <sup>i</sup> Þ consisting of letters N,S, which fixes an order of bundles of O<sup>þ</sup> <sup>i</sup> O� i -curves at a semi-circumvention of the point O<sup>þ</sup> <sup>i</sup> O� i in the circle Ω in the direction of increasing u.

We shall describe a T-type of a point O<sup>þ</sup> <sup>i</sup> O� i with a word A<sup>þ</sup> <sup>i</sup> (A� <sup>i</sup> Þ, and a T-type of a point Oi with words A� i .

T-types of IR points O� <sup>0</sup> ð Þ 0; 0 of Eq. (1) systems are described in the following theorem.

Theorem 2. Let a number u ¼ 0 be the multiplicity k ∈f g 0;…; 3 of the root of a polynomial P uð Þ of the Eq. (1) system. Then, words A� <sup>0</sup> , which determine the topological types (T-types) of IR points O� <sup>0</sup> ð Þ 0; 0 of this system, depending on the value of k and a sign of a number apk (where a and pk are coefficients of the system), have the forms as shown in Table 3 [5].

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

<sup>i</sup> ui ð Þ ; 0 , i∈f1; …; mg.

ui ki gi A<sup>þ</sup>

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.

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

5.2. The double change (DC) transformation

mation in the opposite case.

2 multipliers, respectively:

Table 4. T-types of IR points O�

5.1. Basic concepts and notation

Section 5.

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

+(�) 1, 3 + Nþð Þ N� S�ð Þ S<sup>þ</sup> +(�) 1, 3 \_ S�ð Þ S<sup>þ</sup> Nþð Þ N� +(�)2 + S�Nþð Þ ∅ ∅ð Þ N�S<sup>þ</sup> +(�)2 \_ ∅ð Þ N�S<sup>þ</sup> S�Nþð Þ ∅

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

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).

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 transfor-

PðuÞ∶ ¼ Xð Þ� 1; u p3ð Þ u � u<sup>1</sup> ð Þ u � u<sup>2</sup> ð Þ u � u<sup>3</sup> , Q uð Þ∶ ¼ Yð Þ� 1; u c u � q<sup>1</sup>

Clearly, 10 different types of RSPQ are possible for an arbitrary Eq. (2) system, as C<sup>2</sup>

<sup>X</sup>ð Þ¼ <sup>x</sup>; <sup>y</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> , <sup>Y</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>c</sup> <sup>y</sup> � <sup>q</sup>1<sup>x</sup> <sup>y</sup> � <sup>q</sup>2<sup>x</sup> (2)

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

<sup>i</sup> A�

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

Phase Portraits of Cubic Dynamic Systems in a Poincare Circle

i

73

<sup>5</sup> <sup>¼</sup> <sup>5</sup>!

<sup>3</sup>!2! = 10.

Corollary 4. IR points O� <sup>0</sup> of any Eq:ð Þ1 —system do not have separatrices.

T-types of IR points Oi ui ð Þ ; 0 6¼ Ooð Þ 0; 0 , i ¼ 1,m, of Eq. (1) systems are described in the following theorem.

Theorem 3. Let a real number ui(6¼ 0) be a multiplicity ki∈f1; 2; 3g of the root of a polynomial P uð Þ of an Eq. (1) system. Then for this system, a value gi <sup>=</sup> <sup>P</sup> (ki) (ui)Q(ui) 6¼0 and words A� <sup>i</sup> , which determine topological types (T-types) of IR points O� <sup>i</sup> ui ð Þ ; 0 of this system, depending on the value of ki and signs of numbers ui and gi, have forms as shown in Table 4 [5].

Corollary 5. As can be seen from Theorems 2 and 3, for the IR points of Eq. (1) systems, only a finite number (13) of different T-types are possible. The investigation of these T-types shows that IR-points of each Eq. (1) system have only m separatrices: one separatrice for every singular point Oi ui ð Þ ; 0 , i ¼ 1, m:

Note 3. In Tables 3 and 4, the lower sign index " + " or "–" on every bundle N or S, indicates whether the bundle adjusts to the point O<sup>þ</sup> <sup>i</sup> or to the point O� i from the side u > ui or from the side u < ui of the isocline u ¼ ui.

In Table 3, row 1, a bundle N does not have a lower sign index because as the detailed study of this case shows, it contains O<sup>þ</sup> <sup>i</sup> -curves (O� <sup>i</sup> -curves) in every domain j j u > 0 [5].


Table 3. T-types of IR points O� <sup>0</sup> ð Þ 0; 0 .


Table 4. T-types of IR points O� <sup>i</sup> ui ð Þ ; 0 , i∈f1; …; mg.

3. A notation of a word A<sup>þ</sup>

tion of increasing u.

We shall describe a T-type of a point O<sup>þ</sup>

P uð Þ of the Eq. (1) system. Then, words A�

i .

72 Differential Equations - Theory and Current Research

O<sup>þ</sup> <sup>i</sup> O� i

Oi with words A�

IR points O�

ing theorem.

A�

T-types of IR points O�

Corollary 4. IR points O�

singular point Oi ui ð Þ ; 0 , i ¼ 1, m:

the side u < ui of the isocline u ¼ ui.

this case shows, it contains O<sup>þ</sup>

Table 3. T-types of IR points O�

whether the bundle adjusts to the point O<sup>þ</sup>

<sup>i</sup> (A�


P uð Þ of an Eq. (1) system. Then for this system, a value gi <sup>=</sup> <sup>P</sup> (ki)

<sup>i</sup> , which determine topological types (T-types) of IR points O�

<sup>i</sup> -curves (O�

k apk A<sup>þ</sup>

<sup>0</sup> ð Þ 0; 0 .

<sup>i</sup> O� i

(where a and pk are coefficients of the system), have the forms as shown in Table 3 [5].

Theorem 2. Let a number u ¼ 0 be the multiplicity k ∈f g 0;…; 3 of the root of a polynomial

with a word A<sup>þ</sup>

<sup>0</sup> ð Þ 0; 0 of Eq. (1) systems are described in the following theorem.

<sup>0</sup> ð Þ 0; 0 of this system, depending on the value of k and a sign of a number apk

<sup>0</sup> of any Eq:ð Þ1 —system do not have separatrices.

T-types of IR points Oi ui ð Þ ; 0 6¼ Ooð Þ 0; 0 , i ¼ 1,m, of Eq. (1) systems are described in the follow-

Theorem 3. Let a real number ui(6¼ 0) be a multiplicity ki∈f1; 2; 3g of the root of a polynomial

depending on the value of ki and signs of numbers ui and gi, have forms as shown in Table 4 [5].

Corollary 5. As can be seen from Theorems 2 and 3, for the IR points of Eq. (1) systems, only a finite number (13) of different T-types are possible. The investigation of these T-types shows that IR-points of each Eq. (1) system have only m separatrices: one separatrice for every

Note 3. In Tables 3 and 4, the lower sign index " + " or "–" on every bundle N or S, indicates

In Table 3, row 1, a bundle N does not have a lower sign index because as the detailed study of

0 0 N N 0, 2 + (�) Nþð Þ N� N�ð Þ N<sup>þ</sup> 1, 3 + (�) N�Nþð Þ ∅ ∅ð Þ N�N<sup>þ</sup>

<sup>i</sup> or to the point O�

i

<sup>i</sup> -curves) in every domain j j u > 0 [5].

from the side u > ui or from

<sup>0</sup> A�

0

<sup>i</sup> Þ consisting of letters N,S, which fixes an order of bundles of

in the circle Ω in the direc-

<sup>i</sup> Þ, and a T-type of a point

(ui)Q(ui) 6¼0 and words

<sup>i</sup> ui ð Þ ; 0 of this system,

<sup>i</sup> O� i

<sup>i</sup> (A�

<sup>0</sup> , which determine the topological types (T-types) of
