(The birth of period 3) This is a hard exercise

(The birth of period 3) This is a hard exercise. The goal is to show that the period-3 cycle of the logistic map is born in a tangent bifurcation at r = 1 + 8 =  3.8284  . .... Here are a few vague hints. There are four unknowns: the three period-3 points a, b, c and the bifurcation value r. There are also four equations:  f  ( a ) = b,  f  ( b ) = c,  f  ( c ) = a, and the tangent bifurcation condition. Try to eliminate a, b, c (which we don’t care about anyway) and get an equation for r alone. It may help to shift coordinates so that the map has its maximum at x = 0 rather than x =1/2. Also, you may want to change variables again to symmetric polynomials involving sums of products of a, b, c. See Saha and Strogatz (1995) for one solution, probably not the most elegant one!