Most systems that matter are nonlinear: predators and prey, competing species, a pendulum swinging through large angles, reacting chemicals. Their equations rarely have formula solutions, but the phase plane still works: at each point the right-hand sides give a velocity, trajectories follow it, and they still never cross. What changes is that a nonlinear system can have several equilibria, each with its own local picture, and trajectories that run from one to another.

A landscape of hills and valleys instead of a single bowl or saddle. Water still flows downhill everywhere, but where it ends up depends on which valley it started in.

The pendulum $\theta'=\omega$, $\omega'=-\sin\theta$, angle across and angular velocity up. Near the rest position the trajectories are closed swings; the curves through the upside-down positions $\pm\pi$ separate them from the motions that go over the top.

An autonomous system $x'=f(x,y)$, $y'=g(x,y)$ is nonlinear when or is not a linear combination of and . When , and their partial derivatives are continuous, the existence and uniqueness theorem still holds, so trajectories still never cross and the phase plane is read as in the linear chapter. What is new: several equilibria, closed orbits that are not ellipses, and trajectories that run from one equilibrium to another.

• The pendulum $\theta''=-\sin\theta$ is the system $\theta'=\omega$, , nonlinear because of the sine.

Differentiate the energy along a solution. The chain rule gives omega times its derivative plus the sine times the derivative of theta; substituting the equations, the two terms are equal and opposite. The energy is therefore constant, and every trajectory lies on one of its level curves: closed loops for small energy, the swings, and wavy lines for large energy, the pendulum going over the top.

Kinetic energy plus potential energy, in suitable units.

Differentiate along a solution by the chain rule.

Substitute the two equations.

So each trajectory stays on one level curve of the energy.

A system is nonlinear when a rate involves a product of the unknowns, a power of one, or a function such as a sine of them.

Great work! You now know how to:

• recognise a nonlinear system
• prove that the pendulum conserves its energy
• read swings and motions over the top from the energy
• say what the linear theory keeps and what it loses