# Phase Portraits of Families VII and VIII of the Quadratic Systems
## 3 Proof of Statement (a) of Theorem 1
### The Global Phase Portraits  
The preceding results of the finite and infinite equilibrium points allow 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 \((1-m)^{2}>4\ell n\). We can see that if \(n>0\), then there is a stable hyperbolic node at the origin of chart \(U_{2}\). Since \(b^{2}-4 dn>0\), there exist two real finite equilibrium points, \(p_{+}\) and \(p_{-}\), which are semi-hyperbolic saddle-nodes. Finitely, \((1-m)^{2}>4\ell n\) implies the existence of two infinite equilibrium points in chart \(U_{1}\) (\(P_{+}\) is a hyperbolic saddle and \(P_{-}\) a hyperbolic node). The local phase portraits at all these equilibrium points are shown in Figure 6. The tools for studying the phase portraits were employed for all possible configurations that appear in Figure 7.  
Figure 4: The sequences of blow-ups for obtaining the local phase portrait at the origin of the local \(U_{2}\) chart when \(n=0\), \(b>0\), and \(m>1\). \(n>0\), \(b^{2}-4 dn>0\) and \((1-m)^{2}>4\ell n\) in Figures 7(1-4), but the phase portrait in Figure 7(2) appears by continuity between the phase portraits in Figure 7(1-3);  
\(n>0\), \(b^{2}-4 dn>0\) and \((1-m)^{2}<4\ell n\) from Figure 7(5);  
\(n>0\), \(b^{2}-4 dn>0\) and \((1-m)^{2}=4\ell n\) in Figure 7(6-8);  
\(n>0\), \(b^{2}-4 dn<0\), and \((1-m)^{2}>4\ell n\) in Figure 7(9);  
\(n>0\), \(b^{2}-4 dn<0\), and \((1-m)^{2}<4\ell n\) in Figure 7(10);  
\(n>0\), \(b^{2}-4 dn<0\), \((1-m)^{2}=4\ell n\) in Figure 7(11);  
\(n>0\), \(b^{2}-4 dn=0\), and \((1-m)^{2}>4\ell n\) from Figure 7(12,13);  
\(n>0\), \(b^{2}-4 dn=0\), and \((1-m)^{2}<4\ell n\) in Figure 7(14,15);  
\(n>0\), \(b^{2}-4 dn=0\), \((1-m)^{2}=4\ell n\) in Figure 7(16-20). Phase portraits with \(n<0\) are symmetric with respect to the origins of the coordinates of the preceding eight cases. Figure 5: The distinct topological local phase portraits at the origin of the local \(U_{2}\) chart. Figure 6: The local phase portraits at the finite and infinite equilibrium points for \(n>0\), \(b^{2}-4 dn>0\) and \((1-m)^{2}>4\ell n\). Figure 7: All the distinct topological phase portraits of quadratic system VII. Here, \(s\) (respectively, \(r\)) denotes the number of separatrices of a phase portrait in the Poincaré disc (respectively, canonical regions). 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 symmetric with respect to the origins of the 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 symmetric with respect to the origins of coordinates of all the preceding cases. \(n=0\), \(b=0\), \(d>0\) and \(m<1\) in Figure 7(25); The cases with \(d<0\) are symmetric with respect to the origins of coordinates of all the preceding cases. \(n=0\), \(b=0\), \(d>0\) and \(m=1\) in Figure 7(26);  
\(n=0\), \(b=0\), \(d<0\) and \(m=1\) in Figure 7(27). Of course, from Table 1, the phase portraits with different numbers 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 in Table 1 are also topologically different. Phase portraits 26 and 27 of Figure 7 are topologically different because phase portrait 27 has two elliptic sectors and phase portrait 26 has no elliptic sectors. Phase portraits 11 and 24 of Figure 7 are topologically different because phase portrait 24 has two elliptic sectors and phase portrait 11 has no elliptic sectors. Phase portraits 18 and 22 of Figure 7 are topologically different because phase portrait 18 has orbits starting at the origin of the local \(U_{2}\) chart and ending at the origin of the local \(U_{1}\) chart, and these kinds of orbits do not exist in phase portrait 22.