Editing Logistic map (section)

=== Scaling limit ===
{{Main|Feigenbaum function}}
[[File:Logistic_map_approaching_the_scaling_limit.webm|thumb|478x478px|Approach to the scaling limit as <math>r</math> approaches <math>r^* = 3.5699\cdots</math> from below.]][[File:Logistic iterates, together, r=3.56994567.svg|class=skin-invert-image|thumb|489x489px|At the point of chaos <math>r^* = 3.5699\cdots</math>, as we repeat the period-doublings<math>f^{1}_{r^*}, f^{2}_{r^*}, f^{4}_{r^*}, f^{8}_{r^*}, f^{16}_{r^*}, \dots</math>, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees, converging to a fractal.]]
Looking at the images, one can notice that at the point of chaos <math>r^* = 3.5699\cdots</math>, the curve of <math>f^{\infty}_{r^*}</math> looks like a fractal. Furthermore, as we repeat the period-doublings<math>f^{1}_{r^*}, f^{2}_{r^*}, f^{4}_{r^*}, f^{8}_{r^*}, f^{16}_{r^*}, \dots</math>, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees. 

This suggests to us a scaling limit: if we repeatedly double the function, then scale it up by <math>\alpha </math> for a certain constant <math>\alpha </math>:<math display="block">f(x) \mapsto - \alpha f( f(-x/\alpha ) ) 
 </math>then at the limit, we would end up with a function <math>g </math> that satisfies <math> g(x) = - \alpha g( g(-x/\alpha ) ) </math>. This is a [[Feigenbaum function]], which appears in most period-doubling routes to chaos (thus it is an instance of '''universality'''). Further, as the period-doubling intervals become shorter and shorter, the ratio between two period-doubling intervals converges to a limit, the first Feigenbaum constant <math>\delta = 4.6692016\cdots </math>.[[File:Logistic scaling with varying scaling factor.webm|thumb|480x480px|For the wrong values of scaling factor <math>\alpha </math>, the map does not converge to a limit, but when <math>\alpha = 2.5029\dots </math>, it converges.]]
[[File:Logistic scaling limit, r=3.56994567.svg|class=skin-invert-image|thumb|487x487px|At the point of chaos <math>r^* = 3.5699\cdots</math>, as we repeat the functional equation iteration <math>f(x) \mapsto - \alpha f( f(-x/\alpha ) ) 
 </math> with <math>\alpha = 2.5029\dots </math>, we find that the map does converge to a limit.]]The constant <math>\alpha </math> can be numerically found by trying many possible values. For the wrong values, the map does not converge to a limit, but when it is <math>\alpha = 2.5029\dots </math>, it converges. This is the second Feigenbaum constant.