Period doubling cascades
Consider the function
f(x) = x- (F a+4 x a) tanh(2/3 x)
and consider an interation xn+1 = f(xn).
Below the fixed points and periodic orbits of this map are shown
for F=128, 200 and 450, as a is increased from 0 to 0.25.
F=128
F=200
F=450
Here is a blowup of the start of the cascade in the last picture:
In the limit F -> oo, the upper branch of the above picture
can be described
by a simpler a map. After some rescaling, it looks like
w-> w + A (exp(-w)-1). Its bifurcations are shown below:
