Chapter 10.7, Problem 8E

Interpretation Introduction

Interpretation:

To show that the map $\text{f}\left({\text{x}}_{\text{n}}\text{,r}\right){\text{= -r + x - x}}^{\text{2}}$ undergoes tangential bifurcation at the origin when $\text{x = 0}$.

To show that for small and positive r, it typically takes many iterations to pass through the bottleneck.

To find out how N(r) scales as $r\to 0$; to show that at the origin ${\text{f}}^{\text{2}}$ looks like a rescaled version of f; also to show that it takes approximately $\frac{\text{N(r)}}{\text{2}}$ iterations for orbits of ${\text{f}}^{\text{2}}$ to pass through the bottleneck, where N(r) is number of iterations required to pass through the bottleneck.

By expanding ${\text{f}}^{\text{2}}\left(\text{x,r}\right)$ and neglecting higher order terms, it is to be shown that ${\text{f}}^2\left(\text{X,R}\right)\cong \text{-R + }X{\text{ - X}}^2$, and this renormalization implies that $\frac{\text{N(r)}}{\text{2}}\approx \text{N}\left(\text{4r}\right)$

To show that the equation in (d) has the solution $\text{N(r)=}{\text{ar}}^{\text{b}}$ and solve for b.

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 corresponding values of r and then renormalizing one map into the 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 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 universal scale factor, and $\text{g(x)}$ is defined in terms of itself.