# Phase Portraits of Families VII and VIII of the Quadratic Systems
## 4 Proof of Statement (\(b\)) 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 have 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 (3) in the local \(U_{2}\) chart can be expressed as follows:  
\[ \begin{split}\dot{u}&=-u(n+mu+(b-1)v+\ell u^{2}+ auv+dv^{2})=P(u,v),\\ \dot{v}&=-v(n+mu+bv+\ell u^{2}+auv+dv^{2})=Q(u,v), \end{split}  \tag{8}\]  
so the origin of \(U_{2}\) is an infinite equilibrium point. The eigenvalues of the Jacobian matrix of the system at the origin are \(-n\) with a multiplicity of two. Therefore, the origin is a hyperbolic node, stable if \(n>0\), and unstable if \(n<0\). If \(n=0\), then the Jacobian matrix of the system at the origin is the zero matrix, and we need to make blow-ups in order to study the local phase portrait at the origin of \(U_{2}\). 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 \(u\) is a factor of the polynomial \(\Pi=vP_{2}(u,v)-uQ_{2}(u,v)\), where \(P_{2}(u,v)\) and \(Q_{2}(u,v)\) are the terms of the lowest degrees of \(P(u,v)\) and \(Q(u,v)\); in our case, \(\Pi=uv^{2}\). Thus, \(u=0\) is characteristic direction; consequently, before conducting a vertical blow-up, we must conduct a twist so that \(u=0\) no longer acts as a characteristic direction. We accomplish this through the change of variables \((u,v)\rightarrow(u_{1},v_{1})\), where \(u_{1}=u+v\), \(v_{1}=v\). By making this change of variables, the differential system (8) can be expressed as follows:  
\[ \begin{array}{l}\dot{u_{1}}=-mu_{1}^{2}-v_{1}^{2}+(1-b+m)u_{1}v_{1}+(a-d- \ell)u_{1}v_{1}^{2}-\ell u_{1}^{3}+(2\ell-a)u_{1}^{2}v_{1}\\ \dot{v_{1}}=-v_{1}\big{(}mu_{1}+(b-m)v_{1}+\ell u_{1}^{2}+(a-2\ell)u_{1}v_{1}+ (d-a+\ell)v_{1}^{2}\big{)},\end{array}  \tag{9}\]  
The characteristic directions of this system are given by the polynomial \(\Pi=(u_{1}-v_{1})v_{1}^{2}\), so \(u_{1}=0\) is not a characteristic direction, and 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 (9) becomes  
\[ \begin{array}{l}\dot{u_{2}}=u_{2}^{2}\big{(}-m-\ell u_{2}+(1-b+m)v_{2}+(2 \ell-a)u_{2}v_{2}-v_{2}^{2}+(a-d-\ell)u_{2}v_{2}^{2}\big{)},\\ \dot{v_{2}}=u_{2}(-1+v_{2})v_{2}^{2}.\end{array}  \tag{10}\]  
Now, conducting the rescaling of the time with factor \(u_{2}\), we obtain the system  
\[ \begin{array}{l}\dot{u_{2}}=u_{2}\big{(}-m-\ell u_{2}+(1-b+m)v_{2}+(2\ell-a) u_{2}v_{2}-v_{2}^{2}+(a-d-\ell)u_{2}v_{2}^{2}\big{)},\\ \dot{v_{2}}=(-1+v_{2})v_{2}^{2}.\end{array}  \tag{11}\]  
The equilibrium points of system (11) on \(u_{2}=0\) are \((0,0)\), which is double, and \((0,1)\). The eigenvalues of the Jacobian matrix at \((0,0)\) are \(0\) and \(-m\). Thus, \((0,0)\) is a semi-hyperbolic equilibrium point; by applying Theorem 2.19 of [8] to it, it is a saddle-node. The eigenvalues of the Jacobian matrix at \((0,1)\) are \(1\) and \(-b\). Thus, this equilibrium point is hyperbolic, a saddle if \(b>0\), and an unstable node if \(b<0\); see Figure 8A, when \(n=0\), \(m<0\), and \(b<0\). From Figure 8A, we can see that the local phase portrait at the axis \(u_{2}=0\) of system (10) is given in Figure 8B. Now, going back through the vertical blow-up and 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 (8) in Figure 8C. Finally, ending the twist, we obtain the local phase portrait at the origin of the local \(U_{2}\) chart, which is shown in Figures 8D and 9A. Figure 8: 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 \(m<0\).