mathematics//dynamical systems//equilibrium and stability//trace-determinant plane

The trace-determinant plane is a chart that classifies the equilibrium of a two-state linear system \(\dot x=Ax\) from two numbers of its matrix, the trace and the determinant, and it gives the stability and the shape of the motion of any 2×2 model (a PD loop, two coupled tanks, a pendulum linearized at a point) without computing an eigenvalue. The eigenvalues of a 2×2 matrix are the roots of

The trace-determinant plane is a chart that classifies the equilibrium of a two-state linear system x˙=Ax\dot x=Axx˙=Ax from two numbers of its matrix, the trace and the determinant, and it gives the stability and the shape of the motion of any 2×2 model (a PD loop, two coupled tanks, a pendulum linearized at a point) without computing an eigenvalue. The eigenvalues of a 2×2 matrix are the roots of

λ2−tr⁡(A) λ+det⁡(A)=0,\lambda^2-\operatorname{tr}(A)\,\lambda+\det(A)=0,λ2−tr(A)λ+det(A)=0,

where the trace is the sum of the diagonal and also the sum of the eigenvalues, and the determinant is their product. So the sign of the determinant says whether the two eigenvalues share a sign, the sign of the trace which way they lean, and the discriminant tr⁡2−4det⁡\operatorname{tr}^2-4\dettr2−4det whether they are real or a complex pair.

The plane then reads in four sentences. The equilibrium is stable when the trace is negative and the determinant positive. A negative determinant is a saddle: one direction attracts and the other throws out, as at the top of an inverted pendulum. Above the parabola tr⁡2=4det⁡\operatorname{tr}^2=4\dettr2=4det the eigenvalues are complex and trajectories turn as a spiral, below it they creep straight in or out as a node; on the vertical axis, with zero trace and positive determinant, they circle for ever round a centre.

eigenvalues−0.50 ± 2.00i trace−1.00 determinant4.24 typeStable spiral The system x' = A x with A = [[−0.4, 2.0], [−2.0, −0.6]]: trace −1.00, determinant 4.24, type: stable spiral. Above the parabola with negative trace: a complex pair in the left half, it turns and falls in.

Go through the six presets and watch where the point falls on the plane; from the stable spiral raise d slowly until the trace crosses zero and the spiral opens, then move b and c until the determinant turns negative and a saddle appears.

A loop designer uses it as a sanity check. For two coupled states with decays a,ba,ba,b and cross gains k12,k21k_{12},k_{21}k12​,k21​, the determinant ab−k12k21ab-k_{12}k_{21}ab−k12​k21​ turns negative exactly when the loop gain beats the decays, which is how a reinforcing loop becomes a runaway (feedback loop).

The boundaries are where linearization stops deciding. On the axis det⁡=0\det=0det=0 or tr⁡=0\operatorname{tr}=0tr=0 a nonlinear system can go either way, and the next term of its Taylor polynomial settles it (linearization). Crossing the axis tr⁡=0\operatorname{tr}=0tr=0 as a parameter moves is the birth of an oscillation, a bifurcation.

The chart stops at two states. For a third-order loop the same question is answered by the Routh-Hurwitz conditions on the characteristic polynomial, and in general by the eigenvalues themselves; and a stable point on the chart says nothing about how large the transient gets first (non-normal matrix).