Math Methods · Learn
A differential equation doesn't just have an answer — it has a flow. Every point gets an arrow saying where the system goes next, and the eigenvalues at a fixed point decide whether nearby trajectories spiral in, fly out, or split.
For a planar system $\dot x = f(x,y),\ \dot y = g(x,y)$, attach the velocity vector $(\dot x,\dot y)$ to every point — that's the direction field. A solution is just a curve tangent to the arrows everywhere, so you can read off the qualitative fate of any starting point without a formula.
Near a fixed point, a linear system $\dot{\mathbf{x}} = A\mathbf{x}$ governs the behaviour, and the eigenvalues $\lambda$ of $A$ tell the whole story. With trace $T$ and determinant $D$, $\lambda = \tfrac{T\pm\sqrt{T^2-4D}}{2}$:
| Eigenvalues | Type | Behaviour |
|---|---|---|
| real, opposite signs | saddle | in one way, out another (unstable) |
| real, same sign | node | all in (stable) or all out (unstable) |
| complex, Re ≠ 0 | spiral | spiral in (Re<0) or out (Re>0) |
| pure imaginary | center | closed orbits, neutrally stable |
The rule is simple: a fixed point is stable when every eigenvalue has negative real part. A damped oscillator is a stable spiral; an undamped one (energy conserved) is a center; an inverted pendulum balanced upright is a saddle.
Real systems are nonlinear, and that's where the richness lives. Near each fixed point you can linearize (use the Jacobian) and the eigenvalue rules return. But nonlinear systems also do things linear ones can't: the Van der Pol oscillator pulls every trajectory onto a single self-sustained limit cycle; the pendulum has centers and saddles joined by a separatrix dividing swinging from spinning.
Edit the matrix and watch the classification flip live; click to drop trajectories; switch to the nonlinear presets to see a limit cycle form. Open the lab →
Bjorn Poonen's free MIT 18.03 lecture notes (PDF) cover graphical methods in §31, autonomous equations in §32, and autonomous systems & phase portraits in §33. §37, "How Google search uses an eigenvector," is a fun bonus once you've seen eigenvalues classify a fixed point here — PageRank is the same idea applied to a 4-billion-page matrix.
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