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

Question

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form $\frac{dY}{dt} = AY$ where $A$ is a $2 \times 2$-matrix and $Y$ is the column vector $(x\ y)$.

Answer 1

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!

Answer 2

I find the below caption for the trace-determinant plane a good reminder (this is taken from Hirsch, Smale and Devaney's Differential Equations, Dynamical Systems, and an Introduction to Chaos (3e, p.64)):