For an autonomous system

$$\mathbf x'=\mathbf f(\mathbf x),$$

a phase portrait represents trajectories in state space rather than plotting each state variable separately against time.

An equilibrium $\mathbf x_*$ satisfies

$$\mathbf f(\mathbf x_*)=0.$$

Nearby trajectories reveal its stability. A stable equilibrium keeps sufficiently small perturbations nearby; an asymptotically stable equilibrium also attracts nearby trajectories as time increases.

For a linear system $\mathbf x'=A\mathbf x$, the eigenvalues of $A$ classify common local behaviors. Negative real parts give attraction, positive real parts give instability, while complex eigenvalues can produce spiraling or oscillatory motion.

Phase portraits expose geometry and long-term behavior even when individual trajectories are not written in closed form.