period doubling
A system settles into a steady repeating rhythm: tick, tick, tick, every beat identical. Nudge a parameter and suddenly the beats alternate: strong, weak, strong, weak, so it now takes two beats before the pattern truly repeats. The rhythm's period has doubled. Keep nudging and it doubles again to four, then eight, faster and faster, until repetition dissolves into chaos.
Period doubling is a bifurcation of a discrete map (or of the return map on a Poincaré section of a flow) in which a period-n cycle loses stability and gives birth to a stable period-2n cycle. For a map x_{n+1} = F(x_n), a fixed point (period-1 cycle) is stable while the magnitude of the multiplier F'(x*) is less than 1; the flip bifurcation occurs when this multiplier passes through -1. The negative sign is the essence: the orbit overshoots to the other side each step, so it must visit two distinct values before returning, doubling the period. Analyzing the second-iterate map F(F(x)) turns each such event back into a pitchfork-like split.
An infinite cascade of these doublings, period 2, 4, 8, 16, ..., accumulating at a finite parameter value is one of the universal routes to chaos, seen in the logistic map, dripping faucets, driven pendulums, nonlinear circuits, and Rayleigh-Benard convection. Remarkably, the cascade converges geometrically at a rate governed by the Feigenbaum constant, the same number for a huge class of systems. Beyond the accumulation point lies chaos, but interleaved with narrow windows of restored periodicity.
In the logistic map x_{n+1} = r x_n (1 - x_n), a stable single value gives way to a 2-cycle at r = 3, a 4-cycle at about r = 3.449, an 8-cycle at about r = 3.544, and so on, the doublings piling up at r_infinity = 3.5699..., where chaos begins.
Successive doublings crowd together toward a finite threshold, the doorway to chaos.
Period doubling is not the same as a driven system happening to respond at half the drive frequency; it is an intrinsic loss of stability signaled by a Floquet multiplier (map eigenvalue) passing through -1, and its cascade is quantitatively universal.