A Small Zoo of Equilibria
Guide 1 of this rung taught you to see a planar system x' = f(x, y), y' = g(x, y) as a velocity vector field — an arrow at every point telling a particle where to drift next — and to read a phase portrait as the family of trajectories threading those arrows. Guide 2 zoomed in on an equilibrium point, where both velocities vanish, and showed that near such a point a smooth nonlinear system behaves like a *linear* one, x' = A x. This guide answers the natural next question: when the motion really is linear, what shapes can the portrait near the origin actually take?
The wonderful answer is that there are only a handful. A 2x2 matrix A has two eigenvalues, and those two numbers — real or complex, same sign or opposite, positive or negative — decide everything about the local picture. Out of all that variety, the portraits sort into a tiny zoo of named species: the node, the saddle, the spiral, and the center, plus two rarer borderline animals. Learn to recognize the four main ones and you can name the behaviour of almost any linear equilibrium at a glance.
Two Real Eigenvalues: Nodes and Saddles
Start with the friendly case, two real distinct eigenvalues lambda1 and lambda2. From the eigenvalue method you already know the general solution is x = C1 e^(lambda1 t) v1 + C2 e^(lambda2 t) v2, a blend of two straight-line solutions running along the eigenvector directions v1 and v2. The entire portrait is just this blend, and the *signs* of the two eigenvalues tell you the story. Think of each term as a dial: e^(lambda t) shrinks toward the origin if lambda < 0 and blows up away from it if lambda > 0.
If both eigenvalues are negative, both dials shrink: every trajectory funnels into the origin, and you have a stable [[node|node]] (also called an attracting or stable node). If both are positive, run the film backwards — every path flees outward in an unstable node. There is a subtle but visible detail: trajectories enter (or leave) tangent to the *slow* eigenvector, the one whose eigenvalue is closer to zero, because the faster-decaying term dies first and the slow direction dominates the approach. That tangency is the giveaway that lets you sketch a node correctly rather than as a vague blur.
Now flip one sign: let lambda1 < 0 < lambda2, eigenvalues of opposite sign. Along v1 motion decays toward the origin; along v2 it grows away from it. A particle is pulled in along one axis while being flung out along the other, and the result is the unmistakable [[ode-saddle-point|saddle point]]: trajectories sweep in close, curve, and shoot back out, tracing hyperbola-like arcs. The two special straight lines are the separatrices — the stable one (along v1) and the unstable one (along v2) — and they divide the plane into four regions. A saddle is *always* unstable: almost every nearby start eventually escapes.
Complex Eigenvalues: Spirals and Centers
When the characteristic equation of A has complex eigenvalues lambda = alpha ± i beta, there are no real eigenvector directions to run straight along — and that absence is exactly what produces rotation. The imaginary part beta is the engine of turning: solutions carry factors like e^(alpha t) cos(beta t) and e^(alpha t) sin(beta t), the sine and cosine spinning the state around the origin while the e^(alpha t) envelope grows or shrinks the radius. So the shape is set by one number, the real part alpha.
If alpha < 0, the radius shrinks while the angle keeps turning: trajectories wind inward forever without ever arriving, a stable [[spiral-point|spiral point]] (a spiral sink, also called a focus). If alpha > 0, the same rotation spirals outward — an unstable spiral source. And if alpha = 0 exactly, the eigenvalues are *purely imaginary*, ±i beta: there is rotation but no growth or decay at all. The radius is frozen, so every trajectory is a closed loop endlessly circling the origin. That is a [[center|center]], ringed by closed orbits, and it is the only one of our four species that is neither attracting nor repelling — it is *neutrally stable*.
The Borderline Cases: Stars and Degenerate Nodes
The four main species cover the generic situations. The two leftover animals appear when the eigenvalues are *equal* — a repeated real root lambda — and they split by how many eigenvectors that root supplies. If lambda has two independent eigenvectors (the matrix is a pure scaling, A = lambda I), then *every* direction is a straight-line solution and trajectories are perfect rays through the origin. That is a [[star-node|star node]], stable if lambda < 0 and unstable if lambda > 0.
If instead the repeated root is *defective* — only one eigenvector exists — you meet the [[improper-node|improper node]] (the degenerate node) from the previous rung's repeated-eigenvalue guide. With just one invariant line, trajectories cannot fan along two rays; they all come in (or out) tangent to that single direction, curling as the extra t e^(lambda t) term grows. It is a kind of squashed node, sharing the stability of its eigenvalue's sign. Both star and improper nodes are knife-edge cases: the tiniest change to A can tip them into an ordinary node or a spiral.
eigenvalues of A equilibrium type stability -------------------------- ---------------------- -------------------- real, same sign node sink if <0, source if >0 real, opposite signs saddle always unstable complex, alpha != 0 spiral (focus) sink if alpha<0, source if alpha>0 pure imaginary (alpha = 0) center neutrally stable repeated, 2 eigenvectors star node sink if <0, source if >0 repeated, 1 eigenvector improper (degenerate) sink if <0, source if >0
Reading a Portrait: From Matrix to Sketch
Let us turn the classification into a habit you can run on any 2x2 system. The point is that you never have to fully solve x' = A x to know its shape — the eigenvalues alone carry the verdict, which is why this qualitative reading is so cheap and so powerful. Here is the routine, walked through once.
- Find the eigenvalues. Solve det(A - lambda I) = 0. (The next guide shows a shortcut that skips even this, using only the trace and determinant of A.)
- Classify by their nature: real same-sign means a node, real opposite-sign means a saddle, complex with nonzero real part means a spiral, pure imaginary means a center.
- Decide stability from the signs: negative (or negative real part) attracts, positive repels, a saddle is unstable regardless, a center is neutral.
- Add direction. For real eigenvalues, draw the eigenvector lines and the tangency along the slow one; for complex, test one velocity arrow to fix the sense of rotation.
One honest warning to carry forward. This crisp zoo is a property of *linear* systems; for a nonlinear equilibrium it describes the linearization, and linearization is trustworthy only at hyperbolic points — those whose eigenvalues all have nonzero real part. The center is precisely non-hyperbolic (alpha = 0), so a linear center may, in the true nonlinear system, actually be a slow spiral in disguise; the closed loops are the most fragile prediction we make. Nodes, saddles, and spirals, being hyperbolic, survive the leap to the nonlinear world intact — which is exactly the bridge the next rung on nonlinear stability will build.