nonlinear dynamics

A system is nonlinear when its response is not proportional to its input. Double the push, and you do not get double the result — you might get three times as much, or half, or something qualitatively different. Most of physics is nonlinear. The linear systems that fill introductory textbooks are approximations, valid near equilibrium and nowhere else.

The simple pendulum is the gentlest entry point into nonlinear dynamics. At small angles the restoring force is proportional to displacement, and the motion is a pure sinusoid. But push harder and the sin θ in the equation of motion can no longer be replaced by θ. The period starts to depend on amplitude. The phase portrait develops a separatrix. Push harder still and the pendulum rotates rather than oscillates — a qualitative change that no linear model can produce.

Nonlinear dynamics encompasses chaos (sensitive dependence on initial conditions), solitons (stable wave packets), bifurcations (sudden qualitative changes in behaviour), and turbulence (the unsolved problem). The pendulum, being exactly solvable, is where most physicists meet nonlinearity for the first time.