# Phase Portraits of Families VII and VIII of the Quadratic Systems
## 1 Introduction and Statement of the Main Results  
A _quadratic polynomial differential system_ (or simply, a _quadratic system_) is a differential system of the following form:  
\[\dot{x}=P(x,y),\qquad\dot{y}=Q(x,y), \tag{1}\]  
where \(P\) and \(Q\) are real polynomials in variables \(x\) and \(y\) and the maximum degree of the polynomials \(P\) and \(Q\) is two.  
At the beginning of the 20 th century, the study of quadratic systems began. In [1], [PERSON] noted how [PERSON] [2], in 1904, published the first work on quadratic systems. Two short surveys on quadratic systems were published, i.e., by [PERSON] [1] in 1966 and by [PERSON] and [PERSON] [3] in 1982.  
In recent decades, quadratic systems were intensively studied and many good results were obtained, see references [4; 5; 6]. In the second reference, one can find many applications for quadratic systems. Although quadratic systems have been studied in more than one thousand papers, we do not have a complete understanding of these systems.  
In [7], the authors prove that any quadratic system is affine-equivalent, scaling the time variable, if necessary, to a quadratic system of the form  
\[\dot{x}=P(x,y),\qquad\dot{y}=Q(x,y)=d+ax+by+\ell x^{2}+mxy+ny^{2},\]  
where \(\dot{x}=P(x,y)\) is one of the following ten:  
\begin{tabular}{l l l l} (I) & \(\dot{x}=1+xy\), & (VI) & \(\dot{x}=1+x^{2}\), \\ (II) & \(\dot{x}=xy\), & (VII) & \(\dot{x}=x^{2}\), \\ (III) & \(\dot{x}=y+x^{2}\), & (VIII) & \(\dot{x}=x\), \\ (IV) & \(\dot{x}=y\), & (IX) & \(\dot{x}=1\), \\ (V) & \(\dot{x}=-1+x^{2}\), & (X) & \(\dot{x}=0\). \\ \end{tabular}  
Roughly speaking, the Poincare disc is the disc centered at the origin of \(\mathbb{R}^{2}\) and the radius, where the interior of this disc is identified with the whole plane \(\mathbb{R}^{2}\) and its boundarycircle \(\mathbb{S}^{1}\) is identified with the infinity of the plane, \(\mathbb{R}^{2}\). This is due to the fact that in the plane, we can go to infinity in as many directions as points on the circle \(\mathbb{S}^{1}\). For more details on the Poincare compactification, see Section 2.2; for the definition of topologically equivalent phase portraits in the Poincare disc, see Section 2.3.  
We note that quadratic system \(X\) has straight lines with constant \(x=\) coordinates formed by orbits, and the conic \(Q(x,y)=0\) is filled with equilibrium points, so the phase portraits are trivial. On the other hand, quadratic systems IX does not contain any equilibrium points, thus making this quadratic system a subclass of the so-called chordal quadratic system. The phase portraits of these systems in the Poincare disc have been completely studied in [7]. Thus, the aim of this paper is to classify the different topological phase portraits in the Poincare disc of the classes of quadratic systems VII and VIII, i.e., of systems  
\[\dot{x}=x^{2},\qquad\dot{y}=d+ax+by+\ell x^{2}+mxy+ny^{2}, \tag{2}\]  
and  
\[\dot{x}=x,\qquad\dot{y}=d+ax+by+\ell x^{2}+mxy+ny^{2}, \tag{3}\]  
respectively.  
Our main result is as follows:  
Theorem 1: _The following two statements hold:_  
1. _The family of quadratic systems_ VII _has_ \(27\) _topologically different phase portraits in the Poincare disc._
2. _The family of quadratic systems_ VIII _has_ \(25\) _topologically different phase portraits in the Poincare disc._  
Statements (a) and (b) of Theorem 1 are proved in Sections 3 and 4, respectively.  
The paper is organized as follows. In Section 2, we present the basic results of equilibrium points and the Poincare compactification. In Sections 3 and 4, we first study the local phase portraits of the finite equilibrium points, and then explore the local phase portraits of the infinite equilibrium points. Finally, we analyze the phase portraits of quadratic systems (2) and (3) in the Poincare disc, respectively.