Editing Logistic map (section)

=== Renormalization estimate ===
The Feigenbaum constants can be estimated by a renormalization argument. (Section 10.7,<ref name=":0" />).

By universality, we can use another family of functions that also undergoes repeated period-doubling on its route to chaos, and even though it is not exactly the logistic map, it would still yield the same Feigenbaum constants.

Define the family <math display="block">f_r(x) = -(1+r)x + x^2</math>The family has an equilibrium point at zero, and as <math>r</math> increases, it undergoes period-doubling bifurcation at <math>r = r_0, r_1, r_2, ...</math>.

The first bifurcation occurs at <math>r = r_0 = 0</math>. After the period-doubling bifurcation, we can solve for the period-2 stable orbit by <math>f_r(p) = q, f_r(q) = p</math>, which yields <math display="block">\begin{cases}
p = \frac 12 (r + \sqrt{r(r+4)}) \\
q = \frac 12 (r - \sqrt{r(r+4)})
\end{cases}</math>At some point <math>r = r_1</math>, the period-2 stable orbit undergoes period-doubling bifurcation again, yielding a period-4 stable orbit. In order to find out what the stable orbit is like, we "zoom in" around the region of <math>x = p</math>, using the affine transform <math>T(x) = x/c + p</math>. Now, by routine algebra, we have<math display="block">(T^{-1}\circ f_r^2 \circ T)(x) = -(1+S(r)) x + x^2 + O(x^3)</math>where <math>S(r) = r^2 + 4r - 2, c = r^2 + 4r - 3\sqrt{r(r+4)}</math>. At approximately <math>S(r) = 0</math>, the second bifurcation occurs, thus <math>S(r_1) \approx 0</math>.

By self-similarity, the third bifurcation when <math>S(r) \approx r_1</math>, and so on. Thus we have <math>r_n \approx S(r_{n+1})</math>, or <math>r_{n+1} \approx \sqrt{r_{n}+6}-2</math>. Iterating this map, we find <math>r_\infty = \lim_n r_n \approx \lim_n S^{-n}(0) = \frac 12(\sqrt{17}-3)</math>, and <math>\lim_n \frac{r_\infty - r_n}{r_\infty - r_{n+1}} \approx S'(r_\infty) \approx 1 + \sqrt{17}</math>.

Thus, we have the estimates <math>\delta \approx 1+\sqrt{17} = 5.12...</math>, and <math>\alpha \approx r_\infty^2 +4r_\infty- 3 \sqrt{r_\infty^2+4r_\infty} \approx -2.24...</math>. These are within 10% of the true values.