Editing Logistic map (section)

=== Other scaling limits ===
When <math>r</math> approaches <math>r \approx 3.8494344</math>, we have another period-doubling approach to chaos, but this time with periods 3, 6, 12, ... This again has the same Feigenbaum constants <math>\delta, \alpha</math>. The limit of <math display="inline">f(x) \mapsto - \alpha f( f(-x/\alpha ) ) 
 </math> is also the same [[Feigenbaum function]]. This is an example of '''universality'''.[[File:Logistic_map_approaching_the_period-3_scaling_limit.webm|thumb|482x482px|Logistic map approaching the period-doubling chaos scaling limit <math>r^* = 3.84943\dots</math> from below. At the limit, this has the same shape as that of <math>r^* = 3.5699\cdots</math>, since all period-doubling routes to chaos are the same (universality).]]

We can also consider period-tripling route to chaos by picking a sequence of <math>r_1, r_2, \dots</math> such that <math>r_n</math> is the lowest value in the period-<math>3^n</math> window of the bifurcation diagram. For example, we have <math>r_1 = 3.8284, r_2 = 3.85361, \dots</math>, with the limit <math>r_\infty = 3.854 077 963\dots</math>. This has a different pair of Feigenbaum constants <math>\delta= 55.26\dots, \alpha = 9.277\dots</math>.<ref name=":1">{{Cite journal |last1=Delbourgo |first1=R. |last2=Hart |first2=W. |last3=Kenny |first3=B. G. |date=1985-01-01 |title=Dependence of universal constants upon multiplication period in nonlinear maps |url=https://link.aps.org/doi/10.1103/PhysRevA.31.514 |journal=Physical Review A |language=en |volume=31 |issue=1 |pages=514–516 |doi=10.1103/PhysRevA.31.514 |pmid=9895509 |bibcode=1985PhRvA..31..514D |issn=0556-2791}}</ref> And <math>f^\infty_r</math>converges to the fixed point to <math display="block">f(x) \mapsto - \alpha f(f( f(-x/\alpha ) )) 
 </math>As another example, period-4-pling has a pair of Feigenbaum constants distinct from that of period-doubling, even though period-4-pling is reached by two period-doublings. In detail, define <math>r_1, r_2, \dots</math> such that <math>r_n</math> is the lowest value in the period-<math>4^n</math> window of the bifurcation diagram. Then we have <math>r_1 =3.960102, r_2 = 3.9615554, \dots</math>, with the limit <math>r_\infty = 3.96155658717\dots</math>. This has a different pair of Feigenbaum constants <math>\delta= 981.6\dots, \alpha = 38.82\dots</math>. 

In general, each period-multiplying route to chaos has its own pair of Feigenbaum constants. In fact, there are typically more than one. For example, for period-7-pling, there are at least 9 different pairs of Feigenbaum constants.<ref name=":1" />

Generally, <math display="inline">3\delta \approx 2\alpha^2 </math>, and the relation becomes exact as both numbers increase to infinity: <math>\lim \delta/\alpha^2 = 2/3</math>.