Period doubling bifurcations are evident when we consider an even number of compositions of the quadratic map. In Figure 2.18
Figure 2.18: Second iterate of the quadratic map near a tangency.
we show the second iteration of the quadratic map
near a tangency. Below the period doubling bifurcation, a single stable
period one orbit exists. As 
is increased, the period one orbit
becomes unstable, and a stable period two orbit is born. This information is
summarized in the bifurcation diagram presented in Figure 2.19.
Figure 2.19: Period doubling (flip) bifurcation diagram.
Let
be the parameter value at which the period doubling bifurcation
occurs. At this parameter value the period one and the nascent period two
orbit coincide. As illustrated in Figure 2.18,
Figure 2.20: Tangency mechanism near a period doubling (flip)
bifurcation.
is
tangent to y = x so that 
. However,
that is, at a period doubling bifurcation the function determining
the local stability of the periodic
orbit is always -1.
Figure 2.20 shows 
just after period doubling.
In general, for a period n to period 2n bifurcation,
and
A period doubling bifurcation is also
known as a flip bifurcation .
In the period one to period two bifurcation,
the period two orbit flips from side to side about its period one parent
orbit. This is because 
(see Prob. 2.14). The first flip bifurcation in the
quadratic map occurs at 
and was analyzed
in sections 2.5.2-2.5.4,
where we considered the location and stability of the period one and
period two orbits in the quadratic map.
