Henri Poincaré, 1880s — the way of drawing that states long-run behaviour without solving anything
At the end of the nineteenth century Poincaré was stuck on the three-body problem: sun, earth and moon pulling on each other, equations of motion easy to write, solution impossible to write. There was then — and still is — no formula giving their positions at an arbitrary time.
His pivot changed the discipline: if you cannot solve it, change the question. Instead of "where is the earth in year 500?", ask "will any of these bodies escape?", "does the orbit stay bounded?", "does the arrangement come back around periodically?". None of those needs a closed-form solution. They only need a clear view of roughly where the motion is going.
To get that view he invented this way of drawing. It answers exactly one question: with no solution to the equations, how do you state a system's long-run behaviour?
The whole apparatus has one requirement: the right state variables. Pick the wrong ones, or too few, and the picture you draw will be self-consistent, handsome, and unrelated to the real system.
Below is the full phase portrait of a pendulum with friction. The small grey arrows are the vector field — each one says "in this state, this is where you move next". The coloured curves are two trajectories: two ways of letting go, each following the arrows.
Three things can be read straight off the picture: ① the teal dot in the middle is a stable fixed point (hanging still) and every arrow around it points inward; ② the two hollow dots left and right are unstable fixed points (balanced upright) with arrows spreading away from them; ③ with friction present, every start winds into the middle — so apart from those two isolated upright points, the entire picture is one basin.
The teal curve is worth a second look: it almost cleared the upright point on the right, moved very slowly in that neighbourhood (the arrows there are short), and slid back after all. The closer you are to an unstable fixed point, the slower the system moves — a fact later turned into a whole family of early-warning methods (critical slowing down, Topic 36).
Zoom in on any one fixed point and the arrow pattern comes in only a handful of varieties. Those patterns settle everything the system does nearby:
Telling which one you have means linearising the rule near the fixed point — in plain words, asking two things: after a small displacement, does the restoring force pull you back or push you further, and does the displacement get carried around as well? Pulled back without rotation = node; pulled back with rotation = spiral; pulled back one way and pushed away the other = saddle; rotation with no pull = centre. Mathematically these are four cases for the eigenvalues of the Jacobian, but for use, those four sentences are enough.
The saddle deserves extra attention: it is where separatrices come from. A trajectory arriving exactly along the "pulled back" direction heads into the saddle forever; a trajectory a hair off gets flung along the other direction, to the left or the right depending on which side the hair was on. That is how a boundary between two different endings gets built.
The construction is not itself a theory about the world; it is a language. But being a language is exactly why it translates a pile of unrelated phenomena into the same sentence:
Why some systems return to how they were and others never do. The first have a stable fixed point; the second have a saddle nearby, or no attractor at all. "Why a lake that turned turbid will not clear again" in ecology and "why a state collapses so fast once it starts" in psychology are the same statement: not insufficient force, but a position on the far side of a divide.
Why some oscillations need no external cause. If the portrait contains an attracting closed loop (a limit cycle), the system rises and falls on its own schedule. The heart's sinoatrial node, predator and prey counts, some business-cycle models all belong here. The habit of hunting for external causes keeps coming up empty in this family.
Why "deterministic" and "predictable" are different words. The definition of chaos is written in this language: trajectories confined to a bounded set (a strange attractor), never repeating, with neighbouring trajectories separating exponentially. Without phase-space vocabulary the sentence cannot even be stated.
It also earns its keep in engineering: phase-plane methods in control theory are used directly to judge whether a loop will oscillate or diverge — and the "state machine" every programmer uses is this construction's discrete-world cousin.