Phase Portrait
The qualitative picture of all trajectories in state space, revealing a system's behavior at a glance.
Phase Portrait
A phase portrait is the complete picture of a system's trajectories drawn in its State Space. Instead of plotting one variable against time, you plot the state variables against each other and sketch the family of curves the flow produces. The result is a kind of weather map of the dynamics: from it you can read where motion comes to rest, where it circulates, and which way nearby trajectories drift — all without solving a single equation.
This qualitative, geometric reading of behavior is the heart of the modern theory of dynamical systems, an approach that goes back to Poincaré.
The anatomy of a portrait
A phase portrait is organized around a few landmark features:
- Fixed points — places where the flow stalls, f(x^*) = 0. Each is classified by how trajectories approach or flee it: nodes, saddles, spirals, centers. See Fixed Point.
- Closed orbits — loops that the motion repeats forever. An isolated, attracting loop is a Limit Cycle.
- Separatrices — special trajectories that divide the plane into regions of different fate, typically emanating from a saddle.
Together these skeletal features determine the behavior of every trajectory, because ordinary trajectories must thread between them without crossing.
Read the field
Here is the phase plane of a damped oscillator, \dot x = y,\; \dot y = -x - 0.3y. Notice the inward spiral: the friction term bleeds energy, so every trajectory winds down to the stable spiral fixed point at the origin.
Compare it mentally with the frictionless case \dot y = -x, whose portrait is nested closed circles, and with \dot y = -x + 0.3y, an unstable spiral that throws trajectories outward. Same template, three very different fates — the difference is one coefficient, and the crossover between them is a Bifurcation.
Why it matters
A phase portrait is robust: small changes to the equations usually nudge the picture without reorganizing it. The exceptions — the parameter values where the portrait does reorganize — are precisely the bifurcations, and tracking them is how we understand the onset of oscillation, Chaos, and pattern formation.