JOVANA
Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

The Trace-Determinant Plane

You already know the four faces an equilibrium can wear. The trace-determinant plane is the single map that sorts them all — two numbers read straight off a 2x2 matrix tell you node, saddle, spiral, or center, and how stable it is, with no eigenvalues to compute.

Two numbers instead of two eigenvalues

In the last guide you met the cast of characters an equilibrium can be: a node, an saddle, a spiral, or a center. Each one was identified by working out the eigenvalues of the 2x2 matrix A and reading their signs. That works, but it is fiddly — you solve a quadratic, maybe wade through complex numbers, and only then learn the type. The trace-determinant plane is a shortcut that skips the eigenvalues entirely and reads the answer off two simple numbers.

The two numbers are the trace and the determinant of A. For a matrix A = [a, b; c, d], the trace is T = a + d (just add the diagonal) and the determinant is D = ad - bc. That is all the arithmetic you ever do — no row reduction, no eigenvector hunt. The whole point of this guide is that these two quantities already contain every fact about the type and stability of the equilibrium, because they secretly control the eigenvalues.

Why those two? Because the characteristic equation of any 2x2 matrix can be written using nothing but T and D. Solving det(A - lambda I) = 0 always collapses to lambda^2 - T*lambda + D = 0. The trace and the determinant ARE the coefficients of that quadratic, so they fix its roots — the eigenvalues — completely. Knowing T and D is therefore exactly as informative as knowing both eigenvalues, just packaged more conveniently.

The little quadratic that hides everything

Look hard at lambda^2 - T*lambda + D = 0 and two friendly facts fall out. The sum of the two roots equals T, and the product of the two roots equals D. So the determinant is lambda_1 * lambda_2 and the trace is lambda_1 + lambda_2. These two relations are the entire bridge from the numbers you can compute to the eigenvalues you actually care about.

The third ingredient is the discriminant of the quadratic, D_isc = T^2 - 4D. Its sign decides whether the eigenvalues are real or complex — the very fork that separated straight-line motion from spiralling motion two guides ago. When T^2 - 4D is positive the roots are two distinct real numbers; when it is negative they are a complex conjugate pair a +/- bi; when it is exactly zero the two roots collapse into one repeated value. So the single number T^2 - 4D draws the boundary between nodes-and-saddles on one side and spirals-and-centers on the other.

The map and its regions

Now make a literal picture. Draw a plane with the trace T on the horizontal axis and the determinant D on the vertical axis. Every 2x2 system becomes a single point (T, D) on this map, and which region the point lands in tells you the type at a glance. Two curves carve the plane into territories: the horizontal axis D = 0, and the upward parabola D = T^2/4 (the curve where the discriminant T^2 - 4D vanishes).

                    D (determinant)
                        ^
        spiral source   |   spiral sink         <- above parabola D = T^2/4
         (unstable)     |    (stable)               (complex eigenvalues)
      ..................[CENTER on D-axis].................
          node          |        node            <- between axis & parabola
         source         |        sink               (real, same sign)
       (unstable)       |     (stable)
   ---------------------+------------------------> T (trace)
            SADDLE       (D < 0)      SADDLE       <- whole lower half-plane
          (unstable, real roots of opposite sign)

   parabola  D = T^2/4 : repeated eigenvalue (star / improper node)
   positive D-axis (T=0): center -- pure rotation, neutrally stable
The trace-determinant plane. Up-down position (sign of D and of the discriminant) sets the type; left-right position (sign of T) sets the stability.

Read it from the bottom up. The entire lower half, D < 0, is saddle country — opposite-sign real eigenvalues, always unstable. Above the axis (D > 0) but below the parabola the eigenvalues are real and same-signed, giving a node. Above the parabola the discriminant is negative, the eigenvalues turn complex, and you get a spiral. The single vertical line T = 0 with D > 0 — sitting on the positive D-axis — is the razor-thin home of the center, with its closed orbits and pure rotation. One picture, every case.

Stability lives on the left-right axis

There is a beautifully clean division of labour on this map. The up-down direction tells you the SHAPE of the trajectories (saddle, node, spiral, center) while the left-right direction tells you the STABILITY. Stay in the upper half, D > 0, where the equilibrium is not a saddle, and the sign of the trace alone decides it: T negative means everything decays inward (stable), T positive means everything grows outward (unstable). The dividing wall is the line T = 0.

This matches the eigenvalue story exactly, and now you can see why. The trace is the sum of the eigenvalues, lambda_1 + lambda_2. If both real eigenvalues are negative their sum is negative, so T < 0 — and a stable equilibrium is asymptotically stable. If the eigenvalues are complex a +/- bi, the real part a is half the trace (since their sum is 2a = T), so again T < 0 means the spiral winds inward. Left half of the map: stable. Right half: unstable. The trace is stability wearing a different hat.

  1. Compute D = ad - bc. If D < 0 you are done — it is a saddle, and a saddle is always unstable. No further checks needed.
  2. If D > 0, compute the discriminant T^2 - 4D to find the shape: positive gives a node (real eigenvalues), negative gives a spiral (complex eigenvalues), zero is the borderline star/improper node.
  3. Now read the trace T for stability: T < 0 means stable (winding or settling inward), T > 0 means unstable (flung outward), and T = 0 with D > 0 is the special center.

The boundaries, the center, and an honest warning

The curves themselves are the delicate cases, and they deserve respect. On the parabola D = T^2/4 the discriminant is zero and the two eigenvalues are equal — the repeated-root situation that gives a star node or an improper node, sitting on the knife-edge between a clean node and a spiral. On the positive D-axis (T = 0, D > 0) you have a center, a pure rotation. These borderline points are exactly where small changes in the matrix flip the picture from one type to another, which is why they are drawn as thin curves and not fat regions.

Here is the honest warning, and it is the same caution that has shadowed this whole rung. The trace-determinant plane classifies the LINEAR system x' = A x perfectly and completely. But most real systems are nonlinear, and near an equilibrium you replace them by their linear part — the Jacobian matrix — through linearization. The trace-determinant verdict on that Jacobian is trustworthy for the real nonlinear system ONLY at a hyperbolic equilibrium, one whose eigenvalues have nonzero real part (the Hartman-Grobman theorem). That covers nodes, saddles, and genuine spirals — but pointedly NOT centers.

Step back and the map's beauty is that it compresses a whole guide's worth of casework into one diagram. Two arithmetic operations place any 2x2 system on it, and its location announces the type and the stability before you draw a single trajectory. In the final guide of this rung you will put it to work: it becomes the first move in sketching a full phase portrait by hand, telling you what kind of picture you are about to draw.