Nonlinear Stability Analysis

Differential Equations · Axiom Academy

A pendulum obeys one nonlinear equation everywhere — but near each resting point, its local behavior is decided by a linear approximation and two eigenvalues. A pendulum's motion follows '' + (g/L) sin = 0 — nonlinear, because of that sin term. It has two resting points: hanging straight down and balanced straight up . Linearizing near each one tells you everything about how it actually behaves there. Set a starting angle and let go. Gravity pulls the bob back down through = 0 ; how far you started decides how wild the swing looks. The equation is the same at every angle — but the MOTION isn't. Write the system as . Near an equilibrium, its Jacobian gives the LINEAR system that governs small disturbances. Toggle between the two equilibria and watch where the eigenvalues land. Where the linear approximation breaks down The linearized period never changes with amplitude — but the REAL pendulum's does. Drag the starting angle and watch the two periods pull apart. One recipe, every nonlinear system: find the equilibria , linearize with the Jacobian , read the eigenvalues . It's how the hanging pendulum (a center or a stable spiral) and the balanced-upright pendulum (always a saddle) get classified from the same equation — and the same recipe carries over to circuits, populations, and any other system where "nonlinear" would otherwise mean "no clean answer."

This is the written version of the interactive lesson above. See the full Differential Equations course.