the phase portrait
A single trajectory tells you what happens for one starting point. The phase portrait is the whole gallery: a single picture showing a representative set of trajectories all at once, with little arrows marking which way time flows along each. It is like a weather map of the system — at a glance you see which regions are pulled toward a calm centre, which are flung outward, and which circle endlessly. It is the complete qualitative summary of the system in one drawing.
To build a phase portrait you locate the equilibrium points first (the still spots where nothing moves), classify each one locally (is it a sink, a saddle, a spiral?), then sketch enough trajectories to show how the regions between equilibria connect. The arrows of the velocity field guide the direction; nullclines (curves where one variable momentarily stops changing) help you place the strokes. The result is not one curve but the organised pattern of all of them, fitted together so they never cross.
The phase portrait matters because it answers the questions that really concern us about a system — long-term fate, sensitivity to the starting point, the existence of cycles — without any closed-form solution. Two systems with the same phase portrait behave the same way qualitatively even if their formulas differ wildly. This is why qualitative theory speaks of classifying portraits rather than solving equations.
The damped pendulum has a phase portrait with spiral trajectories winding into the rest state at the bottom, plus saddle points at the upside-down balance positions; nearly every trajectory eventually spirals down to rest.
One drawing captures the fate of every start: most motions spiral down and stop.
A phase portrait shows only a representative few trajectories, not all of them (there are infinitely many). The skill is choosing trajectories that reveal the structure — near each equilibrium and across each separating curve — so the rest can be inferred.