phase portrait

Instead of plotting position against time, plot the system's state against itself, position on one axis, velocity on another, and let each trajectory trace its own curve. The resulting picture, filled with flowing curves and the special points they swirl around, is a phase portrait: a single map of every possible future the system can have, all at once.

The phase portrait is the collection of trajectories of a dynamical system drawn in phase space (or state space), the abstract space whose axes are the variables needed to specify the state. For a system dx/dt = f(x), the vector field f assigns an arrow to every point, and trajectories follow those arrows; no two trajectories ever cross, because the future is uniquely determined by the present. The qualitative anatomy, namely fixed points (nodes, saddles, spirals, centers), closed loops (limit cycles), and the separatrices that divide basins of attraction, captures the long-term behavior without ever solving the equations in closed form.

This geometric, qualitative viewpoint is the heart of nonlinear dynamics, associated with Poincaré: when exact solutions are impossible (as they usually are), you can still read off what the system does by studying the shape of its flow. For a first-order system the phase portrait is a line; a two-variable system gives the familiar phase plane; higher dimensions require sections and projections. A caution: crossing trajectories in a drawing signal an error or a projection from higher dimensions, never a genuine autonomous flow.