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 2 n -cycle is created; it becomes superstable and loses stability in a period doubling bifurcation. Self-similarity is mathematically expressed as comparing f with second iterate f 2 at the corresponding values of r and then renormalizing one map into other. The function f can be renormalized by taking its second iterate, rescaling x → x α , and shifting the value of r to the next superstable value. The functional equation for g(x) is g(x) = αg 2 ( x α ) Here, α is the universal scale factor and g(x) is defined in terms of itself. g(x) is the renormalized function of f(x) with renormalization parameter α , which can be written as f(x,R ∞ ) ≈ αf 2 ( x α ,R ∞ ) As, at the onset of chaos, it isn’t required to shift R to renormalize, the above equation can be written as f(x) ≈ αf 2 ( x α ) This limiting equation is also called g(x). Hence g(x) = αg 2 ( x α ) . Where it can be shown that, α= 1 g ( 1 ) . This function g(x) is often written in terms of the power series solution: g ( x ) =1+c 2 x 2 +c 4 x 4 + ...

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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 2n-cycle is created; it becomes superstable and loses stability in a period doubling bifurcation.

Self-similarity is mathematically expressed as comparing f with second iterate f2 at the corresponding values of r and then renormalizing one map into other.

The function f can be renormalized by taking its second iterate, rescaling x→xα, and shifting the value of r to the next superstable value.

The functional equation for g(x) is

g(x) = αg2(xα)

Here, α is the universal scale factor and g(x) is defined in terms of itself.

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

f(x,R∞)≈αf2(xα,R∞)

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

f(x)≈αf2(xα) This limiting equation is also called g(x).

Hence g(x) = αg2(xα). Where it can be shown that, α=1g(1).

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

g(x)=1+c2x2+c4x4+...

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