The Mathematics
For a two-dimensional autonomous system:
dx/dt = f(x, y)
dy/dt = g(x, y)
we draw the vector field (f, g) at a grid of points and trace integral curves — the paths a particle would follow if it "went with the flow." Fixed points occur where f = g = 0; the Jacobian matrix at those points tells us whether they're stable spirals, saddles, nodes, or centres.
Reading the map
A few patterns to recognise:
- Centre: nested closed orbits (Lotka-Volterra). The system cycles forever at an amplitude set by initial conditions.
- Stable spiral: trajectories spiral inward to a fixed point. The system returns to equilibrium after perturbation.
- Saddle: trajectories approach along one axis and diverge along another. The system is extremely sensitive to the direction of perturbation.
- Limit cycle: an isolated closed orbit that attracts or repels nearby trajectories.
Head back up to the playground and switch between systems. Watch how the same initial-condition grid produces qualitatively different portraits. That's the power of the phase plane — the geometry is the dynamics.
Further reading
- Strogatz, Nonlinear Dynamics and Chaos — the gold standard textbook, especially Chapters 6–7.
- Phase portrait (Wikipedia)
- Lotka, A. J. (1925), Elements of Physical Biology