Period-doubling bifurcation

Summary

A period-doubling bifurcation, or flip bifurcation, happens in a family of maps when the derivative at a fixed point crosses $-1$. The fixed point changes stability, and an orbit of period $2$ is born from it. For flows, the same happens to a periodic orbit whose Floquet multiplier crosses $-1$: a new periodic orbit of about twice the period appears. A sequence of such bifurcations, at parameters that accumulate geometrically, is one of the main routes to chaos, governed by the Feigenbaum constants.