Prigogine & Lefever, 1968 — the minimal toy model of a dissipative structure
Chemistry class teaches that reactions run towards equilibrium: concentrations approach some value monotonically and then stop changing. But from the 1950s onwards, solutions that refused to end that way kept turning up in laboratories — most famously the Belousov–Zhabotinsky reaction (the BZ reaction), in which the colour of a beaker swings back and forth periodically for tens of minutes, like a clock. This was widely held to be impossible, and Belousov's paper was rejected twice.
So the question becomes: what allows a system made only of ordinary chemical reactions to start oscillating on its own, and how few ingredients does it take? The answer Ilya Prigogine and René Lefever gave in Brussels in 1968 was a hypothetical reaction network pared back as far as it would go — later nicknamed the "Brusselator" by John Tyson, after the Oregonator.
It is not an attempt to model the real chemistry of the BZ reaction. It answers a cleaner question: what is the minimum parts list for oscillation?
As equations it is two lines (X and Y are concentrations):
dX/dt = A − (B+1)·X + X²·Y
dY/dt = B·X − X²·Y
Rule 2 — feedstock concentrations pinned from outside — carries all the weight and is the easiest to skim past: it is what keeps the system from ever reaching equilibrium. Somebody outside is supplying reagents and clearing waste, and the bill is being paid continuously. The real BZ reaction eventually falls quiet for exactly this reason: the beaker runs out of reagents, and only then does it truly reach equilibrium.
Fix A, raise B slowly, and you get one clean switch:
For small B, X and Y wobble for a while from any starting point and then settle on a fixed pair of concentrations (X = A, Y = B/A). That is a steady state: material keeps flowing through, but the dials stop moving.
When B crosses 1 + A², that steady state goes unstable. The system no longer settles; it runs around a closed orbit instead, X and Y alternately rising and falling with a stable period and amplitude. In state space this self-sustaining loop is a limit cycle — "limit" meaning that whatever concentrations you start from, you are drawn onto the same loop, with an amplitude set by the equations rather than by how you began.
There is a second way to play it: give X and Y diffusion terms so they can spread out in space. When the two diffuse at different rates (the inhibitor travelling faster than the activator), the system grows stationary stripes in space — Turing patterns. Same reactions: without diffusion you get order in time, with it you get order in space.
The Brusselator's value is not in predicting any particular reaction. It is that it settled a question: sustained oscillation needs no special or mysterious ingredient — ordinary reactions plus "the system is continuously fed" is enough. And it hands over a minimum parts list: one autocatalytic positive feedback, one delayed negative feedback, and an unbroken flow of material.
Read the world with that list and the same skeleton shows up all over: oscillations in glycolysis, the rise and fall of proteins through the cell cycle, predator–prey cycles, the rhythmic firing of cardiac and nerve cells. The chemistry and biology have nothing in common; the three parts do. Which is also its central falsifiable expectation: take the parameter measuring distance from equilibrium and push it, and the steady state should go unstable at a specific threshold rather than becoming gradually shakier.