A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
A fixed point (or equilibrium) of ẋ = f(x) is a point x* where f(x*) = 0 — the system sits still. Stability classifies whether nearby trajectories are attracted to x* (stable), repelled from it (unstable), or exhibit mixed behavior (saddle). For linear systems, the eigenvalues of the coefficient matrix completely determine stability. For nonlinear systems, the eigenvalues of the Jacobian at x* determine local stability, provided no eigenvalue has zero real part.
Every dynamical system ẋ = f(x) has a natural starting point for analysis: find the fixed points where f(x*) = 0, then determine their stability. Fixed points are the simplest possible behavior — nothing moves — and yet they organize the entire phase portrait. The stable fixed points are attractors that capture nearby trajectories. The unstable ones repel. The saddle points, with their stable and unstable manifolds, carve phase space into basins of attraction. Understanding fixed points and their stability tells you the skeleton of the dynamics.
Stability comes in degrees. Lyapunov stability means trajectories that start close to x* stay close forever — they don't wander off, but they don't necessarily converge either. Think of a ball rolling in a perfectly frictionless bowl: it oscillates around the bottom but never settles. Asymptotic stability means trajectories not only stay close but actually converge to x* as time progresses — now there's friction, and the ball settles to rest. Exponential stability is stronger still: the convergence rate is bounded by an exponential decay e-αt. For most purposes in nonlinear dynamics, asymptotic stability is the key notion.
For linear systems ẋ = Ax, the eigenvalues of A tell the complete story. All eigenvalues with negative real parts: asymptotically stable. Any eigenvalue with positive real part: unstable. The classification gives nodes (real eigenvalues, same sign), saddles (real eigenvalues, opposite sign), spirals (complex eigenvalues), and centers (purely imaginary eigenvalues). Your linear algebra background in eigenvalues and eigenvectors directly provides the tools: eigenvectors give the directions of fastest growth or decay, eigenvalues give the rates. What's new in the nonlinear context is that this classification applies only locally, at each fixed point, via the Jacobian — and it can fail at borderline cases.
The borderline cases are where the real part of an eigenvalue is exactly zero. Here the linear approximation is structurally unstable: an arbitrarily small perturbation can change the qualitative behavior. A center (purely imaginary eigenvalues) could become a stable spiral, an unstable spiral, or remain a center depending on the nonlinear terms. A zero eigenvalue signals a potential bifurcation — a qualitative change in the system's behavior as parameters vary. These borderline cases are not pathological exceptions; they are the doorways to the richest phenomena in nonlinear dynamics, including bifurcations, limit cycles, and chaos.