Lyapunov Stability
To prove a system settles without solving its equations, find an energy-like function that is lowest at rest and always decreases along the motion. Like a marble in a bowl with friction, the system can only roll downhill, so it must end up at the bottom. That function is a Lyapunov function.
Click in the phase plane to start a new trajectory.
Lyapunov's method
For a pendulum \(\ddot\theta = -\tfrac{g}{L}\sin\theta - b\dot\theta\), take the energy as the candidate function:
\(V\) is zero only at rest and positive elsewhere, and it never increases when \(b > 0\). So every trajectory near the bottom must settle there.
What to notice
The gray rings are level sets of \(V\). With damping, every trajectory crosses the rings inward and spirals to the center, and the energy plot only goes down. With zero damping, trajectories ride along a ring forever: stable but never settling. Make the damping negative, like a controller that adds energy, and trajectories spiral outward. Lyapunov's method is how engineers prove that nonlinear controllers, such as those for walking or flying robots, are stable.