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 ([math]y=x[/math]).

The simplest locus present in the Feigenbaum diagram is that of the fixed points [math]F[/math] of the logistic map [math]f(x)=px(1-x)[/math], that is, the values ​​of [math]x[/math] such that [math]x=px(1-x)[/math]: this is represented in red (once you click on [i]Show fixed points[/i]). Solve this equation with parameter [math]p[/math] in the unknown [math]x[/math], and enter the formula for the solution (substituting the parameter [math]p[/math] with [math]x[/math] in order to let GeoGebra plot the function) confirming that the locus is the declared one.[br][br]The locus of orbits of order 2 shown in green (click on [i]Show orbits of order 2[/i]) corresponds to pairs of distinct numbers [math]x_A[/math] and [math]x_B[/math] such that [math]x_A=f(x_B)[/math] and [math]x_B=f(x_A)[/math]. Clearly, if [math]f_2[/math] denotes the second iterate map [math]f_2(x)=f(f(x))[/math], then orbits of order 2 give fixed points of [math]f_2[/math]:[br][br][math]f_2(x_A)=f(f(x_A))=f(x_B)=x_A[/math], [math]f_2(x_B)=f(f(x_B))=f(x_A)=x_B[/math][br][br]but fixed points of [math]f_2[/math] need not correspond to orbits of order 2, since they can be fixed points of [math]f[/math]. Solve the parametric equation [math]x=f_2(x)[/math] ([i]Hint: use the above observation and Ruffini's theorem to reduce the order of the equation[/i]) and enter the formulae of the two solution corresponding to orbits of order 2 (substituting [math]x[/math] for [math]p[/math] as before) and confirm that the locus is the declared one.[br][br]The locus of orbits of order 4 shown in orange (click on [i]Show orbits of order 4[/i]) is similarly defined.[br][br]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).