AHL 5.17 (HL)—Phase portraits

A phase portrait plots state variables against one another rather than against time. Arrows or trajectories show how the system evolves, and fixed points are states where all derivatives are zero.

Classify a fixed point by nearby trajectories: convergence suggests stability, while divergence suggests instability. Nullclines where one derivative is zero help locate and interpret the flow without solving every trajectory.

In a predator–prey portrait, a closed orbit can indicate repeating population states; it does not mean the two populations are constant. The axes and direction arrows are essential evidence.

A phase portrait is not a time-series graph. It may hide the speed of motion, and a visually central point is not automatically an equilibrium or stable state.

For $\dot x=ax+by$, $\dot y=cx+dy$ with distinct non-zero eigenvalues: positive real parts move away from the origin; negative real parts move toward it; complex values spiral; purely imaginary values give circles or ellipses; real values of opposite signs give a saddle. Exact solutions are required only for distinct real eigenvalues, using eigenvector modes.