{\color{red}y_{\infty,1}} =
{\color{blue}y_{\infty,2}} =
The vector field
F: \mathbb R^2 \to \mathbb R^2,
\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}\mapsto \begin{pmatrix} y_1^2 - y_2^2 \\
Ay_1 + By_1 y_2 \end{pmatrix}
defines a system y' = F(y).
Determine the fixed points
y_{\infty} = \left({\color{red}y_{\infty,1}},{\color{blue}y_{\infty,2}}\right) \neq (0,0).
{\color{red}y_{\infty,1}} =
{\color{blue}y_{\infty,2}} =
{\color{red}y_{\infty,1}} =
{\color{blue}y_{\infty,2}}=
The fixed points are the solutions to the equation F(y) =\begin{pmatrix} 0 \\ 0 \end{pmatrix}.
The first equation is y_1^2 - y_2^2 = (y_1 - y_2) (y_1 + y_2) = 0 and therefore
y_1 = \pm y_2.
The second equation is written
0= Ay_1 + By_1 y_2 = Ay_1 (1 + fractionReduce(B,A) y_2) and
with y_1 \neq 0 follows y_2 = fractionReduce(-A,B).
Thus, we have two fixed points
\displaystyle
y_{\infty} = \left({\color{red}y_{\infty,1}},{\color{blue}y_{\infty,2}}\right)
= \left(\mp fractionReduce(A,B), fractionReduce(-A,B)\right).