Cubic Map Problem

Consider the cubic map x_p+1 = f(x_n) where f(x_n) = rx_n - x^3 _n Find the fixed points. For which values of r do they exist? For which values are they stable? To find the 2-cycles of the map, suppose that f(p) = q and f(q) = p. Show that p, q are roots of the equation x(x^2 -r+ 1)(x^2 - r - 1)(x^4 - rx^2 +1) = 0 and use this to find all the 2-cycles. Determine the stability of the 2-cycles as a function of r. Plot a partial bifurcation diagram, based on the information obtained