Lotka-Volterra predator-prey cycles. Each closed orbit is a different initial population pair.
The Core Idea
A phase portrait is a map of everywhere a dynamical system can go. Instead of plotting a variable against time — position versus t, population versus t — we plot variables against each other: predator vs prey, voltage vs current, x-velocity vs y-velocity.
Every point on the plane is a possible state; the arrows and trajectories show where the system flows from there. The shape of the phase portrait tells you everything about long-term behaviour — stability, oscillation, chaos — without solving a single equation.
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.
Slower speed reveals fine structure. Near the fixed point, orbits are nearly circular.
Lotka-Volterra: Nature's oscillator
The classic predator-prey model:
dx/dt = αx − βxy (prey: grows, eaten)
dy/dt = δxy − γy (predator: fed, dies)
The phase portrait reveals closed orbits around a centre — populations cycle endlessly: prey booms, predators follow, prey crashes, predators starve, repeat. No external forcing needed; the oscillation is intrinsic to the coupling.
Van der Pol limit cycle. All trajectories converge to the same orbit regardless of initial conditions.
Van der Pol: Limit cycles
Switch to the Van der Pol system and you'll see something radically different: trajectories spiral inward from any starting point and lock onto a single closed curve — a limit cycle. This is the mathematical heartbeat of oscillators, from vacuum-tube circuits to pacemaker neurons.
x' = y
y' = μ(1 − x²)y − x
The parameter μ controls nonlinearity. Large μ produces relaxation oscillations (sharp jumps followed by slow drifts); small μ gives nearly sinusoidal motion.
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