Read a system's entire fate from the eigenvalues of one matrix. Give me a 2×2 Jacobian and I place its fixed point in the trace–determinant plane — node, saddle, spiral, or center — and prove the label against the actual eigenvalues, then draw the flow. Rendered, not quoted.
source Strogatz, S.H. (1994), Nonlinear Dynamics and Chaos, Perseus Books — Ch. 5, the trace–determinant classification (after Poincaré's phase-plane analysis). No stable open link; cited author/title/year. catalog
Linearize any planar flow at a fixed point and you get ẋ = A x with the Jacobian A = [[a,b],[c,d]]. Two numbers decide everything:
T = tr A = a+d D = det A = ad−bc
Eigenvalues solve λ² − Tλ + D = 0, so λ = (T ± √(T²−4D))/2. The discriminant Δ = T²−4D splits real from complex.
Rules (exact): D<0 → saddle. D>0, Δ>0 → node. D>0, Δ<0 → spiral. T=0, D>0 → center. Sign of T gives stability (T<0 stable).
An assumption, marked plainly: linearization reads only the hyperbolic cases faithfully. On the borders (T=0, Δ=0, D=0) the nonlinear terms can decide — the center especially is non-generic. amber
A system's fate in two numbers — Poincaré's phase plane. This is the geometry that lives beneath its neighbours: when a parameter drags (T,D) across a boundary of this plane, the portrait changes qualitatively.
Cross into D<0 and a node becomes a saddle — that is the‑bifurcation. Cross the T=0 axis with D>0 and a stable spiral loses stability, birthing a limit cycle — that is the‑hopf‑bifurcation. This sphere is the map they both move on.
Live re-check: does the engine's label still agree with the true eigenvalues of the tampered test case?
Confirms on load. If window 6 swaps in the determinant-only rule, the label stops matching the eigenvalues and this badge flips red.
The four entries of a Jacobian A. Pick a preset fixed point or type your own.
Vector field ẋ=Ax, integrated trajectories (RK4), and — when real — the invariant eigen-directions.
Proven: the classification is exact from (T,D) and matches the eigenvalue signs on all six canonical systems — stable/unstable node, saddle, stable/unstable spiral, center. Eigenvectors verified invariant (Av=λv); a stable node/spiral has both eigenvalues with negative real part, so every trajectory flows in.
Δ=0 (repeated eigenvalue) you get degenerate/star nodes; on D=0 a whole line of fixed points; and the center (T=0) is structurally unstable — the linearization's closed orbits can be an artifact, with the true nonlinear flow spiralling in or out (Hartman–Grobman is silent here).
So the honest engine flags line and center as the fragile cases rather than pretending they are robust.
"Negative determinant just means unstable." → D<0 is a saddle — it has one stable and one unstable direction; some trajectories approach before leaving.
"A spiral and a node are the same if both decay." → They differ by Δ: a node has real eigenvalues (straight-in), a spiral has complex ones (rotating-in). Same stability, different geometry.
"The trace alone tells you the type." → T fixes only stability; the type needs D and Δ. Dropping them is exactly the planted void in window 6.
Planted void (disclosed): classify by the determinant alone — ignore T and Δ. Then a stable and an unstable node look identical, and a spiral is mislabelled a "node." The Witness (7) catches it live.