Why regimes go round in circles
Democracies decay and so does whatever replaces them; I show the loop Polybius described in 150 BC is what the mathematics predicts once a country's birth rate falls fast enough, not a run of bad luck.
How to read the figure
The wobble. Think of a country's politics as a wobble. Parties swap, crises arrive, protests happen, and normally things settle back down. The blue line is that wobble over decades. A wide loop means big political swings; a tight loop means calm.
Why it climbs. The line rises because the birth rate is falling, the only thing changing here. Fewer children means the average voter keeps getting older, and older voters want different things from a constitution than younger ones.
Below the dashed square the loop shrinks to nothing: a crisis hits, the system absorbs it, life returns to normal. That is a constitution doing its job.
Above it the loop widens and never closes. A crisis no longer fades. The country swings from one kind of government to the next and keeps going round. Rome did this. So has most of Latin America.
Test it. Press Shock it near the bottom and the wobble dies. Press it near the top and it never does. That difference is the whole argument.
The technical version
The normal form of a Hopf bifurcation, integrated live. Horizontal axes are the two state variables; the vertical axis is θ, the speed of the demographic transition, drifting upward so the system passes through the bifurcation rather than sitting at a fixed value. For θ < 0 the origin is a stable spiral. At θ = 0 a pair of eigenvalues crosses the imaginary axis, the origin loses stability and a limit cycle of radius √θ appears. The trajectory settles onto that radius only after a lag, the delay effect characteristic of slow passage through a bifurcation. Tested against V-Dem.