1. Introduction and Statement of the Main Results

2. Preliminary Definitions

3. Proof of Statement $\left(\mathit{a}\right)$ of Theorem 1

3.4. The Global Phase Portaits

The preceding results for the finite and infinite singular points, allow to obtain the global phase portraits quite easily, taking into account that the straight line $x=0$ is invariant.

First we consider the case satisfying the following conditions: $n>0$, $b^2-4dn>0$ and ${(1-m)}^2>4\ell n$. We have seen that if $n>0$ then there is a stable hyperbolic node at the origin of the chart $U_2$. Since $b^2-4dn>0$ there exist two real finite singular points $p_{+}$ and $p_{-}$ that are semi-hyperbolic saddle-nodes. Finitely since ${(1-m)}^2>4\ell n$ imply the existence of two infinite singular points in the chart $U_1$ ($P_{+}$ is a hyperbolic saddle and $P_{-}$ a hyperbolic node). The local phase portraits at all these singular points are shown in Figure 6. The tools for studying the phase portraits in this case are employed for all possible configurations which appears in Figure 7.

$n>0$, $b^2-4dn>0$ and ${(1-m)}^2>4\ell n$ in Figure 7(1) to Figure 7(4), but the phase portrait in Figure 7(2) appears by continuity between the phase portraits in Figure 7(1) to Figure 7(3);

$n>0$, $b^2-4dn>0$ and ${(1-m)}^2<4\ell n$ from Figure 7(5);

$n>0$, $b^2-4dn>0$ and ${(1-m)}^2=4\ell n$ in Figure 7(6) to Figure 7(8);

$n>0$, $b^2-4dn<0$, and ${(1-m)}^2>4\ell n$ in Figure 7(9);

$n>0$, $b^2-4dn<0$, and ${(1-m)}^2<4\ell n$ in Figure 7(10);

$n>0$, $b^2-4dn<0$, ${(1-m)}^2=4\ell n$ in Figure 7(11);

$n>0$, $b^2-4dn=0$, and ${(1-m)}^2>4\ell n$ from Figure 7(12) and Figure 7(13);

$n>0$, $b^2-4dn=0$, and ${(1-m)}^2<4\ell n$ in Figure 7(14) and Figure 7(15);

$n>0$, $b^2-4dn=0$, ${(1-m)}^2=4\ell n$ in Figure 7(16) to Figure 7(20).

The phase portraits with $n<0$ are symmetric with respect to the origin of coordinates of the preceding eight cases.

Now we study the phase portraits when $n=0$.

$n=0$, $b>0$ and $m>1$ in Figure 7(21);

$n=0$, $b>0$ and $m<1$ in Figure 7(22);

$n=0$, $b>0$ and $m=1$ in Figure 7(23);

The cases with $b<0$ are the symmetric with respect to the origin of coordinates of the preceding three cases.

$n=0$, $b=0$, $d>0$ and $m>1$ in Figure 7(24);

The cases with $d<0$ are the symmetric with respect to the origin of coordinates of all the preceding case.

$n=0$, $b=0$, $d>0$ and $m<1$ in Figure 7(25);

The cases with $d<0$ are the symmetric with respect to the origin of coordinates of all the preceding case.

$n=0$, $b=0$, $d>0$ and $m=1$ in Figure 7(26);

$n=0$, $b=0$, $d<0$ and $m=1$ in Figure 7(27).

Of course from the Table 1 the phase portraits with different number of separatrices and canonical regions are topologically distinct. Now we shall see that the phase portraits with the same number of separatrices and canonical regions of the Table 1 also are topologically different.

Table 1. Here p.p. denotes phase portrait in the Poincaré disc, s denotes the number of separaratrices of the phase portrait, and r denotes the number of canonical regions of the phase portrait.

Table 1. Here p.p. denotes phase portrait in the Poincaré disc, s denotes the number of separaratrices of the phase portrait, and r denotes the number of canonical regions of the phase portrait.

s	4	5	6	8	9	9	10	11	12
r	1	2	2	3	4	2	3	4	3
p.p.	10	26,27	14	15	23	25	11,24	5	18,22
s	13	14	14	16	17	18	19	20	21
r	4	3	5	5	6	5	6	5	6
p.p.	16,19,20	9	17,21	8	6,7	13	12	2	1,3,4

The phase portraits 26 and 27 of Figure 7 are topologically different because the phase portrait 27 has two elliptic sectors and the phase portrait 26 has no elliptic sectors.

The phase portraits 11 and 24 of Figure 7 are topologically different because the phase portrait 24 has two elliptic sectors and the phase portrait 11 has no elliptic sectors.

The phase portraits 18 and 22 of Figure 7 are topologically different because the phase portrait 18 have orbits starting at the origin of the local chart $U_2$ and ending at the origin of the local chart $U_1$, and this kind of orbits do not exist in the phase portrait 22.

The phase portraits 16, 19 and 20 of Figure 7 are topologically different. First the phase portrait 16 have orbits starting at the origin of the local chart $U_2$ and ending at the origin of the local chart $U_1$, and this kind of orbits do not exist in the phase portrait 19 and 20. The phase portrait 19 has a separatrix starting at the origin of the local chart $U_2$ and ending at an infinite equilibrium point in the local chart $V_1$, and such a kind of separatrix does not exist in the phase portrait 20.

The phase portraits 17 and 21 of Figure 7 are topologically different because the phase portrait 21 has two elliptic sectors and the phase portrait 17 has no elliptic sectors.

The phase portraits 1, 3 and 4 of Figure 7 are topologically different because the unstable separatrix of the lower equilibrium point on the straight line $x=0$ contained in $x>0$ has different ending infinite equilibrium point in the these three phase portraits.

4. Proof of Statement $\left(\mathit{b}\right)$ Theorem 1

4.4. The Global Phase Portaits

The preceding results for the finite and infinite singular points, allow to obtain the global phase portraits quite easily, taking into account that the straight line $x=0$ is invariant.

First we consider the case satisfying the following conditions: $n>0$, $b^2-4dn>0$ and $m^2>4\ell n$. We have seen that $n>0$ denotes a stable hyperbolic node at the origin of the chart $U_2$, $b^2-4dn>0$ indicates the existence of two real finite singular points ($p_{+}$ a hyperbolic unstable node and $p_{-}$ a hyperbolic saddle), and $m^2>4\ell n$ imply two infinite singular points in the chart $U_1$ ($P_{+}$ and $P_{-}$ which are nilpotent saddle-nodes). The local phase portraits at all these singular points are shown in Figure 10.

With the help of Mathematica we have proved that in order that the conditions $n>0$, $b^2-4dn>0$ and $m^2>4\ell n$ hold, the parameters of the differential system (3) must satisfy one of the conditions:

(i)

$b<0$, $d\le 0$, $\ell <0$ and $n>0$;

(ii)

$b<0$, $d\le 0$, $\ell \ge 0$, $n>0$ and $m<-2\sqrt{\ell n}$;

(iii)

$b<0$, $d\le 0$, $\ell \ge 0$, $n>0$ and $m>-2\sqrt{\ell n}$;

(iv)

$b<0$, $d>0$, $\ell <0$ and $0<n<b^2/\left(4d\right)$;

(v)

$b<0$, $d>0$, $\ell \ge 0$, $0<n<b^2/\left(4d\right)$ and $m<-2\sqrt{\ell n}$;

(vi)

$b<0$, $d>0$, $\ell \ge 0$, $0<n<b^2/\left(4d\right)$ and $m>-2\sqrt{\ell n}$;

(vii)

$b=0$, $d<0$, $\ell <0$ and $n>0$;

(viii)

$b=0$, $d<0$, $\ell \ge 0$, $n>0$ and $m<-2\sqrt{\ell n}$;

(ix)

$b=0$, $d<0$, $\ell \ge 0$, $n>0)$ and $m>-2\sqrt{\ell n}$;

(x)

$b>0$, $d\le 0$, $\ell <0$ and $n>0$;

(xi)

$b>0$, $d\le 0$, $\ell \ge 0$, $n>0$ and $m<-2\sqrt{\ell n}$;

(xii)

$b>0$, $d\le 0$, $\ell \ge 0$, $n>0$ and $m>-2\sqrt{\ell n}$;

(xiii)

$b>0$, $d>0$, $\ell <0$ and $0<n<b^2/\left(4d\right)$;

(xiv)

$b>0$, $d>0$, $\ell \ge 0$, $0<n<b^2/\left(4d\right)$ and $m<-2\sqrt{\ell n}$;

(xv)

$b>0$, $d>0$, $\ell \ge 0$, $0<n<b^2/\left(4d\right)$ and $m>-2\sqrt{\ell n}$.

We have proved that in the cases (i), (ii), (iv), (vii) and from (ix) to (xv) we get the phase portrait (1) of Figure 11; in the cases (iii), (vi) and (viii) we obtain the phase portrait (2) of Figure 11; and finally in the case (v) we get the phase portrait symetric to the phase portrait (2) with respect to the straigt line $x=0$. For instance: the phase portrait (1) of Figure 11 is obtained when the parameters of system (3) are $d=a=0$, $b=-1$, $\ell =-1$, $m=-3$ and $n=1$; the phase portrait (2) of Figure 11 is obtained when the parameters are $d=a=0$, $b=-1$, $\ell =1$, $m=3$ and $n=1$.The phase portrait (3) of Figure 11 exists by continuity going from the phase portrait (1) to the phase portrait (2).

We recall that the separatrices of a polynomial differential system in the Poincaré disc are all the orbits at infinity, the finite singularities and the two orbits at the boundary of an hyperbolic sector. Also the limit cycles are separatrices but the quadratic system VIII has no limit cycles. If in a phase portrait of the Poincaré disc we remove all the separatrices the open components which remain are called the the canonical regions of the phase portrait. For more details on the separatrices and canonical regions see [10,11].

The tools for studying the phase portraits of system (3) for the case $n>0$, $b^2-4dn>0$ and $m^2>4\ell n$ are used in the following cases, leading to the next results:

$n>0$, $b^2-4dn>0$, and $m^2<4\ell n$ in Figure 11(4);

$n>0$, $b^2-4dn>0$, $m^2=4\ell n$ and $2an+(1-b)m>0$ in Figure 11(5);

$n>0$, $b^2-4dn>0$, $m^2=4\ell n$ and $2an+(1-b)m<0$ in this case the phase portrait is symmetric with respect to the straight line $x=0$ of the phase portrait of the previous case;

$n>0$, $b^2-4dn>0$, $m^2=4\ell n$ and $2an+(1-b)m=0$ in Figure 11(6);

$n>0$, $b^2-4dn<0$, and $m^2>4\ell n$ in Figure 11(7);

$n>0$, $b^2-4dn<0$, and $m^2<4\ell n$ in Figure 11(8);

$n>0$, $b^2-4dn<0$, $m^2=4\ell n$ and $2an+(1-b)m>0$ in Figure 11(9);

$n>0$, $b^2-4dn<0$, $m^2=4\ell n$ and $2an+(1-b)m<0$ this case is a symetric phase portrait with respect to the straight line $x=0$ of the previous phase phase portrait;

$n>0$, $b^2-4dn<0$, $m^2=4\ell n$ and $2an+(1-b)m=0$ in Figure 11(10) andFigure 11(11);

$n>0$, $b^2-4dn=0$, and $m^2>4\ell n$ from Figure 11(12) to Figure 11(14);

$n>0$, $b^2-4dn=0$, and $m^2<4\ell n$ in Figure 11(15);

$n>0$$b^2-4dn=0$, $m^2=4\ell n$ and $2an+(1-b)m>0$ in Figure 11(16) and Figure 11(17);

$n>0$, $b^2-4dn=0$, $m^2=4\ell n$ and $2an+(1-b)m=0$ in Figure 11(18);

The cases with $n<0$ are the symetric with respect to the straight line $y=0$ to all the preceding cases;

$n=0$, $m>0,$$b>0$ in Figure 11(19);

$n=0$, $m>0,$$b<0$ in Figure 11(20);

$n=0$, $m>0,$$b=0$ and $d>0$ in Figure 11(21);

$n=0$, $m>0,$$b=0$ and $d<0$ this case has a symmetric phase portrait with respect to $y=0$ to the previous case;

The phase portraits of the cases $n=0$ and $m<0$ are symmetric with respect to the straight line $x=0$ of the phase portraits of the cases $n=0$ and $m>0;$

$n=0$, $m=0,$$b>2+l$ and $\ell >0$ in Figure 11(22);

$n=0$, $m=0,$$b>2+l$ and $\ell <0$ this case has a symetric phase portrait with respect to the $y=0$ axis;

$n=0$, $m=0,$$0<b\le 2+\ell$ and $\ell \ne 0$ in Figure 11(23);

$n=0$, $m=0,$$b<0$ and $\ell >0$ in Figure 11(24);

$n=0$, $m=0,$$b<0$ and $\ell <0$ the phase portrait of this case is symetric with respect to the straight line $y=0$ of the previous phase portrait;

$n=0$, $m=0,$$b=0$, $\ell >0$ and $d<0$ in the Figure 11(25);

$n=0$, $m=0,$$b=0$, $\ell <0$ and $d>0$ the phase portrait of this case is symetric with respect to the straight line $y=0$ of the previous phase portrait;

$n=0$, $m=0,$$b=0$, $\ell >0$ and $d>0$ this case has the same phase portrait of Figure 11(8);

$n=0$, $m=0,$$b=0$, $\ell <0$ and $d<0$ this case has the symetric phase portrait with respect to the straight line $y=0$ to the phase portrait of Figure 11(8).

Of course from the Table 2 the phase portraits with different number of separatrices and canonical regions are topologically distinct. Now we shall see that the phase portraits with the same number of separatrices and canonical regions of the Table 2 also are topologically different.

The phase portraits 4 and 10 of Figure 12 are topologically different because the phase portrait 4 has two finite singular points and the phase portrait 10 has no finite singular points.

Table 2. Here p.p. denotes phase portrait in the Poincaré disc, s denotes the number of separaratrices of the phase portrait, and r denotes the number of canonical regions of the phase portrait.

Table 2. Here p.p. denotes phase portrait in the Poincaré disc, s denotes the number of separaratrices of the phase portrait, and r denotes the number of canonical regions of the phase portrait.

s	4	5	6	7	8	8	9	9	10	11
r	1	2	1	2	1	3	2	4	3	4
p.p.	8	25	23	22	11	15	9	24	4,10	21
s	12	13	14	14	15	16	17	18	19	20
r	3	4	3	5	4	5	4	5	4	5
p.p.	17,18	16,19	7	20	5	6	14	12,13	3	1,2

The phase portraits 17 and 18 (respectively 16 and 19) of Figure 12 are topologically different because the phase portrait 17 (respectively 16) has two orbits going to the origin of the chart $U_2$, and such orbits do not exist in the phase portrait 18 (respectively 19).

The phase portrait 14 of Figure 12 has three pairs of infinite singular points while the phase portrats 16 and 19 only have only two pairs of infinite singumlar, so the phase portrait 14 is different from the phase portraits 16 and 19.

We note that the phase portrait 13 of Figure 12 has three pairs of infinite singular points, while the phase portrait 20 has only two pairs, so these two phases portraits are topologically disctinct.

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