The Feigenbaum loci

This applet shows various loci in the Feigenbaum diagram of the logistic map. The blue oblique line is the bisector of the first and third quadrants ().
The simplest locus present in the Feigenbaum diagram is that of the fixed points of the logistic map , that is, the values ​​of such that : this is represented in red (once you click on Show fixed points). Solve this equation with parameter in the unknown , and enter the formula for the solution (substituting the parameter with in order to let GeoGebra plot the function) confirming that the locus is the declared one. The locus of orbits of order 2 shown in green (click on Show orbits of order 2) corresponds to pairs of distinct numbers and such that and . Clearly, if denotes the second iterate map , then orbits of order 2 give fixed points of : , but fixed points of need not correspond to orbits of order 2, since they can be fixed points of . Solve the parametric equation (Hint: use the above observation and Ruffini's theorem to reduce the order of the equation) and enter the formulae of the two solution corresponding to orbits of order 2 (substituting for as before) and confirm that the locus is the declared one. The locus of orbits of order 4 shown in orange (click on Show orbits of order 4) is similarly defined. The bifurcations in the Feigenbaum diagram correspond to passage of stability from orbits of a given order to orbits of double order (fixed points are orbits of order 1).