3. Topological type (T-type) of a singular point O(0, 0)

In order to find all O-curves and to split their totality into the bundles N, S, let us use the method of exceptional directions of a system in the point O [1]. According to this method, the equation of exceptional directions for the point O of the Eq. (1) system has the form.

2) in the case of d ¼ 0 depending on signs of values q and P(q), they have forms, indicated in a

q P (q) AO BO + + <sup>S</sup>0SþNþS<sup>0</sup> <sup>H</sup><sup>2</sup>

\_ \_ <sup>S</sup>0NþSþS<sup>0</sup> PH<sup>2</sup>

0 + <sup>S</sup>0SþNS� <sup>H</sup><sup>2</sup>

) means a bundle S, adjoining to point O(0,0) from the domain x > 0 along a semi-axis

whether the bundle consists of Oþ-curves or of O�-curves. Upper index 1 or 2 on every such a bundle indicates whether its O-curves are adjoining to point O along a straight line y ¼ q1x or

In Table 2, row 5, 6, a bundle N does not have a lower sign index because it contains both O<sup>þ</sup> -

Corollary 1. From Theorem 1, it follows, that Eq. (1) systems do not have limit cycles on the

Indeed, such a cycle could surround a singular point O (0,0) of an Eq. (1) system, and then the Poincare index of this singular point must be equal to 1 [1]. However, Bendixon's formula for

where e hð Þ is the number of elliptical (hyperbolic) O-sectors of the system. This formula and our Theorem 1 give: for the singular point O (0, 0) of every Eq. (1) system, Poincare index

Corollary 2. For the singular point O (0, 0) of an Eq. (1) system, 11 different topological types

e � h 2

the index of an isolated singular point of a smooth dynamic system is as follows:

(T-types) are possible, and from the analysis of these 11 T-types we can conclude:

I Oð Þ¼ 1 þ

Note 2. Let us clarify the meaning of the new symbols introduced in Theorem 1.

The lower sign index " + " or "–" on every bundle N or S, different from S<sup>0</sup> and S<sup>0</sup>

x ¼ 0, y < 0, when t ! þ∞ (along a semi-axis x ¼ 0, y > 0, when t! �∞).

, BO = HH (Table 1).

, indicates

P

69

P

P

<sup>S</sup>�N� <sup>H</sup><sup>2</sup>

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<sup>N</sup>�S� PH<sup>2</sup>

<sup>S</sup>� PH<sup>2</sup>

Table 2,

S<sup>0</sup> (S<sup>0</sup>

R2

x;y plane.

I(O) = 0.

along a straight line y ¼ q2x:

curves and O� -curves simultaneously.

3) in the case of d < 0 they have forms: AO = S0S<sup>0</sup>

Table 2. T-type of the singular point O(0, 0) when d = 0.

+ \_ S0S<sup>0</sup>

\_ + S0S<sup>0</sup>

0 \_ NSþS<sup>0</sup>

$$x\mathcal{Y}(x,y) \equiv x(a\mathbf{x}^2 + b\mathbf{x}y + cy^2) = 0.$$

For this, the following cases are possible:

1. When <sup>d</sup> � <sup>b</sup><sup>2</sup> � <sup>4</sup>ac <sup>&</sup>gt; <sup>0</sup>, this equation defines simple straight lines <sup>x</sup> <sup>¼</sup> 0 and.

$$y = q\_i \mathbf{x}\_{\prime} \quad i = 1, 2, \ q\_1 < q\_2$$

2. When d ¼ 0, this equation defines the straight line x ¼ 0 and the double straight line.

$$y = q\mathbf{x}, q = -\frac{b}{2c}$$

3. When d < 0, the equation defines only the straight line x ¼ 0:

Theorem 1 is true for the aforementioned cases [5].

Theorem 1. Words AO and BO, which define a topological type (T-type) of a singular point O (0, 0) of the Eq. (1) system:

1) in the case of d > 0, depending on signs of values Pðqi Þ ¼ 1, 2, have forms, indicated in a Table 1;


Table 1. Т-type of a singular point when d > 0 (r = 1, 6).


Table 2. T-type of the singular point O(0, 0) when d = 0.

1) T-type of a singular point O in its form BO is easy to construct using its Т-type in the form

2) Real roots of a polynomial P(u) (polynomial Q(u)) are in fact angular coefficients of isoclines

3) When we write out the real roots of the system's polynomials P(u), Q(u), separately or all

In order to find all O-curves and to split their totality into the bundles N, S, let us use the method of exceptional directions of a system in the point O [1]. According to this method, the

xY xð Þ� ; <sup>y</sup> x ax<sup>2</sup> <sup>þ</sup> bxy <sup>þ</sup> cy<sup>2</sup> <sup>¼</sup> <sup>0</sup>:

x, i ¼ 1, 2, q<sup>1</sup> < q<sup>2</sup>

2c

þN<sup>2</sup> þS0 N1 �S2 � <sup>¼</sup> <sup>S</sup>0S<sup>1</sup>

þS2 þS0 S1 �N<sup>2</sup> � <sup>¼</sup> NS<sup>2</sup> þS0 S1

þN<sup>2</sup> þS0 S1 �S2

þS2 þS0 N1 �N<sup>2</sup>

Þ ¼ 1, 2, have forms, indicated in a

� PH<sup>2</sup>

<sup>þ</sup> PH<sup>2</sup>

PEP

� PEPH<sup>3</sup>

� <sup>H</sup><sup>3</sup>

þNS<sup>2</sup>

2. When d ¼ 0, this equation defines the straight line x ¼ 0 and the double straight line.

<sup>y</sup> <sup>¼</sup> qx, q ¼ � <sup>b</sup>

Theorem 1. Words AO and BO, which define a topological type (T-type) of a singular point O

r P (q1) P (q2) AO BO

equation of exceptional directions for the point O of the Eq. (1) system has the form.

1. When <sup>d</sup> � <sup>b</sup><sup>2</sup> � <sup>4</sup>ac <sup>&</sup>gt; <sup>0</sup>, this equation defines simple straight lines <sup>x</sup> <sup>¼</sup> 0 and.

y ¼ qi

3. When d < 0, the equation defines only the straight line x ¼ 0:

Theorem 1 is true for the aforementioned cases [5].

1) in the case of d > 0, depending on signs of values Pðqi

1, 4 + + S0S<sup>1</sup>

2\_ \_ S0N<sup>1</sup>

3, 6 \_ + S0N<sup>1</sup>

5+ \_ S0S<sup>1</sup>

Table 1. Т-type of a singular point when d > 0 (r = 1, 6).

(0, 0) of the Eq. (1) system:

Table 1;

AO, and going backward (we need to determine both forms, see Corollary 1);

together, we always number the roots of each one of them in an ascending order.

3. Topological type (T-type) of a singular point O(0, 0)

of infinity (isoclines of a zero));

68 Differential Equations - Theory and Current Research

For this, the following cases are possible:

2) in the case of d ¼ 0 depending on signs of values q and P(q), they have forms, indicated in a Table 2,

3) in the case of d < 0 they have forms: AO = S0S<sup>0</sup> , BO = HH (Table 1).

Note 2. Let us clarify the meaning of the new symbols introduced in Theorem 1.

S<sup>0</sup> (S<sup>0</sup> ) means a bundle S, adjoining to point O(0,0) from the domain x > 0 along a semi-axis x ¼ 0, y < 0, when t ! þ∞ (along a semi-axis x ¼ 0, y > 0, when t! �∞).

The lower sign index " + " or "–" on every bundle N or S, different from S<sup>0</sup> and S<sup>0</sup> , indicates whether the bundle consists of Oþ-curves or of O�-curves. Upper index 1 or 2 on every such a bundle indicates whether its O-curves are adjoining to point O along a straight line y ¼ q1x or along a straight line y ¼ q2x:

In Table 2, row 5, 6, a bundle N does not have a lower sign index because it contains both O<sup>þ</sup> curves and O� -curves simultaneously.

Corollary 1. From Theorem 1, it follows, that Eq. (1) systems do not have limit cycles on the R2 x;y plane.

Indeed, such a cycle could surround a singular point O (0,0) of an Eq. (1) system, and then the Poincare index of this singular point must be equal to 1 [1]. However, Bendixon's formula for the index of an isolated singular point of a smooth dynamic system is as follows:

$$I(O) = 1 + \frac{e - h}{2}$$

where e hð Þ is the number of elliptical (hyperbolic) O-sectors of the system. This formula and our Theorem 1 give: for the singular point O (0, 0) of every Eq. (1) system, Poincare index I(O) = 0.

Corollary 2. For the singular point O (0, 0) of an Eq. (1) system, 11 different topological types (T-types) are possible, and from the analysis of these 11 T-types we can conclude:

for every Eq. (1) system, the singular point O(0, 0) has not more than four separatrices (actually 2, 3, or 4 ones).

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

dv

a system, which in the coordinates τ, v, z looks like the following:

one of the IR-points of the first kind, namely with the point <sup>1</sup>

system. Consequently, the following corollary is correct.

the first kind.

are displayed in a circle Ω, filling it.

boundary Γ of a circle Ω, filling it.

∀i ∈ 1;…; mg for the point O<sup>þ</sup>

<sup>i</sup> O� i

1. Let a O<sup>þ</sup>

call trajectories of the Eq. (1) system on Γ.

O�

ordinary point p∈ Ω and adjacent to a point Oþ �ð Þ

2. A notation for bundles N, S, adjacent to a point O<sup>þ</sup>

notation introduced for the point O (0, 0).

<sup>i</sup> ui ð Þ ; 0 : O<sup>þ</sup>

<sup>i</sup> O� i

<sup>d</sup><sup>τ</sup> ¼ �X vð Þþ ; <sup>1</sup> Y vð Þ ; <sup>1</sup> vz,

x,y onto a Poincare sphere ∑ with the diametrically

Phase Portraits of Cubic Dynamic Systems in a Poincare Circle

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71

:

opposite points identified, considered without its equator. Every Eq. (1) system transforms into

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

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)

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

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))

2) Such a system's infinitely remote trajectories (including IR points) are displayed on the

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

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.

� �∈ Γþ �ð Þ ≔ Γ

� �—curve be a semi-trajectory of the Eq. (1) system in Ω, starting in an

<sup>i</sup> :

� � <sup>x</sup>><sup>0</sup> ð Þ <sup>x</sup><<sup>0</sup> :

<sup>i</sup> O� i

� � from the circle Ω, similar to the

<sup>i</sup> O� i

� �, � we shall introduce the following notation.

dz

<sup>d</sup><sup>τ</sup> <sup>¼</sup> Y vð Þ ; <sup>1</sup> <sup>z</sup><sup>2</sup>

<sup>v</sup><sup>0</sup> ; 0 � �,
