Chapter 10.7, Problem 4E

Interpretation Introduction

Interpretation:

To verify the following statements:Function g(x) near the origin is roughly parabolic.

The function g(x) has infinitely many wiggles as x ranges over the real line.

Concept Introduction:

Renormalization is based on the self-similarity of the Figtree. The twigs look like the earlier branches, except they are scaled down in both x and r directions. The Figtree structure shows an endless repetition of the same dynamical processes, a ${\text{2}}^{\text{n}}\text{-cycle}$ is created; it becomes superstable and loses stability in a period doubling bifurcation.

Self-similarity is mathematically expressed as comparing $\text{f}$ with second iterate ${\text{f}}^{\text{2}}$ at the corresponding values of r and then renormalizing one map into other.

The function $\text{f}$ can be renormalized by taking its second iterate, rescaling $\text{x}\to \frac{\text{x}}{\text{α}}$, and shifting the value of r to the next superstable value.

The functional equation for $\text{g(x)}$ is

${\text{g(x) = αg}}^{\text{2}}\left(\frac{\text{x}}{\text{α}}\right)$

Here, $\text{α}$ is the universal scale factor and $\text{g(x)}$ is defined in terms of itself.

g(x) is the renormalized function of f(x) with renormalization parameter $\text{α}$, which can be written as

${\text{f(x,R}}_{\infty }\text{)}\approx {\text{αf}}^{\text{2}}\left(\frac{\text{x}}{\text{α}}{\text{,R}}_{\infty }\right)$

As, at the onset of chaos, it isn’t required to shift R to renormalize, the above equation can be written as

$\text{f(x)}\approx {\text{αf}}^{\text{2}}\left(\frac{\text{x}}{\text{α}}\right)$ This limiting equation is also called g(x).

Hence ${\text{g(x) = αg}}^{\text{2}}\left(\frac{\text{x}}{\text{α}}\right)$. Where it can be shown that, $\text{α=}\frac{\text{1}}{\text{g}\left(\text{1}\right)}$.

This function g(x) is often written in terms of the power series solution:

$\text{g}\left(\text{x}\right){\text{=1+c}}_{\text{2}}{\text{x}}^{\text{2}}{\text{+c}}_{\text{4}}{\text{x}}^{\text{4}}\text{+}\mathrm{...}$