# Phase Portraits of Families VII and VIII of the Quadratic Systems
## 4 Proof of Statement (\(b\)) Theorem 1
### The Global Phase Portraits  
The preceding results for the finite and infinite equilibrium points allowed us 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: \(n>0\), \(b^{2}-4 dn>0\), and \(m^{2}>4\ell n\). We have seen that \(n>0\) denotes a stable hyperbolic node at the origin of chart \(U_{2}\), \(b^{2}-4 dn>0\) indicates the existence of two real finite equilibrium points (\(p_{+}\), which is a hyperbolic unstable node, and \(p_{-}\), which is a hyperbolic saddle), and \(m^{2}>4\ell n\) implies two infinite equilibrium points in chart \(U_{1}\) (\(P_{+}\) and \(P_{-}\), which are nilpotent saddle-nodes). The local phase portraits at all these equilibrium points are shown in Figure 10.  
With the help of Mathematica, we proved that in order for the conditions \(n>0\), \(b^{2}-4 dn>0\), and \(m^{2}>4\ell n\) to hold, the parameters of the differential system (3) must satisfy one of the following conditions:  
(i) \(b<0\), \(d\leq 0\), \(\ell<0\) and \(n>0\);  
(ii) \(b<0\), \(d  
(2), with respect to the straight line \(x=0\). For instance, 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 \(n=1\); phase portrait (2) of Figure 11 is obtained when the parameters are \(d=a=0\), \(b=-1\), \(\ell=1\), \(m=3\), and \(n=1\). Phase portrait (3) of Figure 11 exists by continuity, from phase portrait (1) to phase portrait (2). We recall that the _separatrices_ of a polynomial's differential system in the Poincare disc are all orbits at infinity, the finite equilibria, and the two orbits at the boundary of a hyperbolic sector. Also, the limit cycles are separatrices but quadratic system VIII has no limit cycles. In a phase portrait of the Poincare disc, if we remove all separatrices, the open components that remain are called the _canonical regions_ of the phase portrait. For more details on the separatrices and canonical regions, see [11; 12]. The tools used for studying the phase portraits of system (3) for \(n>0\), \(b^{2}-4 dn>0\) and \(m^{2}>4\ell n\) are used in the following cases, leading to the following results:  
\(n>0\), \(b^{2}-4 dn>0\), and \(m^{2}<4\ell n\) in Figure 11(4);  
\(n>0\), \(b^{2}-4 dn>0\), \(m^{2}=4\ell n\), and \(2 an+(1-b)m>0\) in Figure 11(5);  
\(n>0\), \(b^{2}-4 dn>0\), \(m^{2}=4\ell n\), and \(2 an+(1-b)m<0\), in this case, the phase portrait is symmetric with respect to the straight line \(x=0\) of the phase portrait of the previous case;  
\(n>0\), \(b^{2}-4 dn>0\), \(m^{2}=4\ell n\), and \(2 an+(1-b)m=0\) in Figure 11(6);  
\(n>0\), \(b^{2}-4 dn<0\), and \(m^{2}>4\ell n\) in Figure 11(7);  
\(n>0\), \(b^{2}-4 dn<0\), and \(m^{2}<4\ell n\) in Figure 11(8);  
\(n>0\), \(b^{2}-4 dn<0\), \(m^{2}=4\ell n\), and \(2 an+(1-b)m>0\) in Figure 11(9);  
\(n>0\), \(b^{2}-4 dn<0\), \(m^{2}=4\ell n\), and \(2 an+(1-b)m<0\); this case is a symmetric phase portrait with respect to the straight line \(x=0\) of the previous phase phase portrait;  
\(n>0\), \(b^{2}-4 dn<0\), \(m^{2}=4\ell n\), and \(2 an+(1-b)m=0\) in Figure 11(10,11);  
\(n>0\), \(b^{2}-4 dn=0\), and \(m^{2}>4\ell n\) from Figure 11(12-14);  
\(n>0\), \(b^{2}-4 dn=0\), and \(m^{2}<4\ell n\) in Figure 11(15);  
\(n>0\)\(b^{2}-4 dn=0\), \(m^{2}=4\ell n\), and \(2 an+(1-b)m>0\) in Figure 11(16,17);  
\(n>0\), \(b^{2}-4 dn=0\), \(m^{2}=4\ell n\), and \(2 an+(1-b)m=0\) in Figure 11(18); The cases with \(n<0\) are symmetric with respect to the straight line \(y=0\) in all 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 Figure 11: All distinct topological phase portraits of quadratic system VIII. Here, \(s\) (respectively, \(r\)) denotes the number of separatrices of a phase portrait in the Poincaré disc (respectively, canonical regions).