I'm surprised that nobody has yet mentioned the logistic map f(x) = kx (1-x) where k in (0, 4] is a parameter that controls how chaotic the solutions are.

For k<=1 the only fixed point is 0; for 1 < k <= 3 there is another fixed point at 1 -

^{1}/

_{k}; for 3 < k <= 1+sqrt(6) (approx. 3.4495) f

^{n}(x) converges to a 2-cycle; for successively smaller intervals we get a 4-cycle, 8-cycle, 16-cycle, etc. and enter a chaotic region above around k = 3.57. (See the Wikipedia entry at

http://en.wikipedia.org/wiki/Logistic_map for more detail and a nice bifurcation diagram.)

Obviously this doesn't fit the problem as stated since f(0) = f(1) = 0, but it should be possible to twerk it slightly and still keep most of the interesting behaviour, e.g. f(x) = k(x+d) (1+d-x) where we make d small (and positive), and limit k so that f(

^{1}/

_{2}) = k(

^{1}/

_{4} + d + d

^{2}) < 1. 0 is not a stable fixed point of the original map for k > 1, so you should be able to get the same sort of results.