I am struggling on how to classify equilibria
The system I am working with is:
$\dot{x}=y$
$\dot{y}=-ay-k^2sinx$
where $a>0$
I managed to find the equilibria as $(0,0), (\pi,0),(-\pi,0)$ as well as all the other multiples of $\pi$
I got the Jacobian for $(0,0)$ to be
$$\begin{pmatrix} 0&1\\ -k^2&-a \end{pmatrix}$$
with eigenvalues $$\lambda=\frac{-a\pm \sqrt{a^2-4k^2}}{2}$$
I got the Jacobian for $(\pi,0)$ to be $$\begin{pmatrix} 0&1\\ k^2&-a \end{pmatrix}$$ with eigenvalues $$\lambda=\frac{-a\pm \sqrt{a^2+4k^2}}{2}$$
I guess what I am asking is have I got it right so far, and if so, how would I go about classifying these when all I know is that $a>0$ ?
In the first case, as long as $0<2k\le a$, both eigenvalues are negative, for $a<2k$ they have negative real part. So it is always a sink, a node in the first case, a spiral in the second, with an improper node in the middle.
For the second case, note that $-a-\sqrt{a^2+4k^2}<0$ and $-a+\sqrt{a^2+4k^2}=\frac{4k^2}{a+\sqrt{a^2+4k^2}}>0$, so that you have a saddle point.