The trace-determinant plane, classification of equilibria of differential equations

Stable solutions only occur when the trace $\tau \leq 0$ and the determinant $\Delta>0$.

So saddles occur when $\Delta<0$, and the non-isolated fixed points whenever $\Delta=0$ (the borderline).

When $\Delta>0$ the centers (these should seem interesting) occur at $\tau=0$.

After that, for $\Delta>0$, just try to recognize the symmetry of stability reversal on either side of the $\tau=0$ line. For example as you go away from the centers, you get spirals, then the borderline stars and degenerate nodes on the $\tau^2-4\Delta=0$ curve, and then the nodes.

I think I accidentally memorized this plane by solving examples of every type of equation that could possibly happen. Very useful for stability analysis by eye!

There are a few concepts to consider:

First of all you need to rebember that these such systems can be solved by writing down the eigenvalues (and theirs eingenvectors) to the plane, the eingenvectors show the diretion of expansion (or retraction) of the system, while the signal of the eingenvalues decide if it is a retraction or a expansion.

I usually reccor to these materials every time I need to remember these concepts in a quick way:

Youtube ODE ovewview

Trace-determinant plane

PS.: Without time to go deeper in this issue, I should came back to review in a feew weeks years.