I would like to calculate the range of values for which x* (fixed point) = 0 and I want to investigate the type of stability.
$$ \frac{d}{du} \begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}-5&a\\2&1\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix}$$
I have already satisfied the cases for node and saddle.
I want to find the solution for a focus:
From my understanding:
Focus = eigenvalues which are complex conjugate numbers with a positive real part.
In my solution I have obtained that the eigenvalues are:
$-2 \pm \sqrt{9+2a}$
However I am unsure of how to satisfy the values for which the stability point is a focus.
Any help would be welcome.
As @Moo pointed out you need to have complex conjugate eigenvalues. That implies that the expression under the square root has to be negative. Hence,
$$9+2a<0 \implies a<-9/2 \implies a < -4.5.$$
In this case, your eigenvalues will be
$$\lambda_{1/2}=-2\pm i\,\sqrt{|2a+9|}$$
which corresponds to a stable focus.