the trace-determinant plane

Classifying a linear equilibrium can feel like memorising a long checklist of eigenvalue cases. The trace-determinant plane folds that whole checklist into one master map. You compute just two numbers from the system's 2-by-2 matrix A — its trace T (the sum of the diagonal entries) and its determinant D — plot the point (T, D), and read the equilibrium's type straight off where the point lands. One picture organises every possibility.

It works because the two eigenvalues of a 2-by-2 matrix are completely captured by T and D: they satisfy lambda^2 - T lambda + D = 0, with sum T and product D. So the map divides up like this. If D is negative, the eigenvalues are real with opposite signs: a saddle (the entire lower half-plane). If D is positive, look at the discriminant T^2 - 4D: where it is positive the eigenvalues are real and same-signed, giving a node; where it is negative they are complex, giving a spiral; the parabola T^2 = 4D between them is the borderline of improper/star nodes. Within those regions the sign of T sets stability: T negative pulls things in (stable, left side), T positive pushes them out (unstable, right side). The positive D-axis, where T = 0, is the line of centers.

This single diagram is the payoff of the whole linear classification: from two easy arithmetic quantities you instantly name the equilibrium — saddle, stable/unstable node, stable/unstable spiral, or center — and judge its stability, without ever finding the eigenvalues explicitly. It also makes vivid how types border one another, so you can see at a glance how an equilibrium would change type as parameters move the point (T, D) across the map.