# Phase Portraits of Families VII and VIII of the Quadratic Systems
## 3 Proof of Statement (a) of Theorem 1
### The Infinite Equilibrium Point at the Origin of Chart \(U_{2}\)  
Studying the infinite equilibrium points in the local \(U_{1}\) chart, we also studied the infinite equilibrium points in the local \(V_{1}\) chart. Thus, we must see whether the origins of the local \(U_{2}\) and \(V_{2}\) charts are infinite equilibrium points or not. System (2) in the local \(U_{2}\) chart can be expressed as follows:  
\[\dot{u} =-u(n+(m-1)u+bv+\ell u^{2}+auv+dv^{2})=P(u,v), \tag{5}\] \[\dot{v} =-v(n+mu+bv+\ell u^{2}+auv+dv^{2})=Q(u,v),\]  
so the origin of \(U_{2}\) is an infinite equilibrium point. The eigenvalues of the Jacobian matrix of system (5) at the origin are \(-n\) with a multiplicity of two. Therefore, the origin is a hyperbolic stable node if \(n>0\), and an unstable node if \(n<0\). If \(n=0\), then the Jacobian matrix of the system at the origin of the local \(U_{2}\) chart is the zero matrix, and we need to make blow-ups in order to study its local phase portrait. Before conducting a vertical blow-up, we need to be sure that \(u=0\) is not a characteristic direction. If \(u=0\) is a characteristic direction, then \(u\) would be a factor of the polynomial \(\Pi=vP_{2}(u,v)-uQ_{2}(u,v)\), where \(P_{2}(u,v)\) and \(Q_{2}(u,v)\) represent the terms of degree two in \(P(u,v)\) and \(Q(u,v)\). In our case, \(\Pi=u^{2}v\). Thus, \(u=0\) is a characteristic direction and, consequently, before conducting a vertical blow-up, we must perform a twist so \(u=0\) no longer acts as a characteristic direction. This is conducted through the change of variables \((u,v)\rightarrow(u_{1},v_{1})\), where \(u_{1}=u+v\), \(v_{1}=v\). By conducting this change of variables, the differential system (5) can be expressed as follows:  
\[\dot{u_{1}} =(1-m)u_{1}^{2}-(b+2-m)u_{1}v_{1}+v_{1}^{2}-\ell u_{1}^{3}-(a-2 \ell)u_{1}^{2}v_{1}+(a-d-\ell)u_{1}v_{1}^{2} \tag{6}\] \[v_{1} =-mu_{1}v_{1}+(m-b)v_{1}^{2}-\ell u_{1}^{2}v_{1}+(2\ell-a)u_{1}v_ {1}^{2}+(a-d-\ell)v_{1}^{3},\]  
Since \(u_{1}=0\) is not a characteristic direction, we can conduct a vertical blow-up. This vertical blow-up is given by the change of variables \((u_{1},v_{1})\rightarrow(u_{2},v_{2})\), where \(u_{2}=u_{1}\), \(v_{2}=v_{1}/u_{1}\). Then, system (6) becomes  
\[\dot{u_{2}} =u_{2}^{2}\big{(}m-1+\ell u_{2}+(b-m+2)v_{2}+(a-2\ell)u_{2}v_{2} -v^{2}-(a-d-\ell)u_{2}v_{2}^{2}\big{)},\] \[v_{2} =-u_{2}v_{2}(-1+v_{2})^{2}.\]  
Now, conducting the rescaling of the time with factor \(u_{2}\), we obtain the system  
\[\dot{u_{2}} =u_{2}\big{(}m-1+\ell u_{2}+(b-m+2)v_{2}+(a-2\ell)u_{2}v_{2}-v^{2 }-(a-d-\ell)u_{2}v_{2}^{2}\big{)},\] \[\dot{v_{2}} =-v_{2}(-1+v_{2})^{2}.\]  
The equilibrium points of the previous system on \(u_{2}=0\) are \((0,0)\) and \((0,1)\) (this is double). The eigenvalues of the Jacobian matrix at \((0,0)\) are \(-1\) and \(1-m\). Thus, the point \((0,0)\) is a hyperbolic stable node if \(m>1\), a hyperbolic saddle if \(m<1\), and for \(m=1\)a semi-hyperbolic saddle-node, according to Theorem 2.19 in [8]. The eigenvalues of the Jacobian matrix at \((0,1)\) are \(0\) and \(-b\). Thus, the local phase portrait of the origin of the local \(U_{2}\) chart is shown in Figure 4A when \(n=0\), \(b>0\), and \(m>1\). Starting from Figure 4A, we obtain the local phase portrait at the axis \(u_{2}=0\) of system (10); see Figure 4B. Going back through the vertical blow, taking into account the value of \(\dot{u}_{1}|_{u_{1}=0}=v_{1}^{2}\), we obtain the local phase portrait at the origin of system (5) in Figure 4C. Finally, undoing the twist, we obtain the local phase portrait at the origin of the local \(U_{2}\) chart, which is shown in Figures 4D and 5A.