To show that the map f ( x n ,r ) = -r + x - x 2 undergoes tangential bifurcation at the origin when 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 → 0 ; to show that at the origin f 2 looks like a rescaled version of f; also to show that it takes approximately N(r) 2 iterations for orbits of f 2 to pass through the bottleneck, where N(r) is number of iterations required to pass through the bottleneck. By expanding f 2 ( x,r ) and neglecting higher order terms, it is to be shown that f 2 ( X,R ) ≅ -R + X - X 2 , and this renormalization implies that N(r) 2 ≈ N ( 4r ) To show that the equation in (d) has the solution N(r)= ar 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 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 corresponding values of r and then renormalizing one map into the other. The function f can be renormalized by taking its second iterate, rescaling x → x α , and shifting the value of r to next superstable value. The functional equation for g(x) is g(x) = αg 2 ( x α ) Here, α is universal scale factor, and g(x) is defined in terms of itself.

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Chapter 10.7, Problem 8E

Interpretation Introduction

Interpretation:

To show that the map f(xn,r)= -r + x - x2 undergoes tangential bifurcation at the origin when 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→0; to show that at the origin f2 looks like a rescaled version of f; also to show that it takes approximately N(r)2 iterations for orbits of f2 to pass through the bottleneck, where N(r) is number of iterations required to pass through the bottleneck.

By expanding f2(x,r) and neglecting higher order terms, it is to be shown that f2(X,R)≅ -R + X - X2, and this renormalization implies that N(r)2≈N(4r)

To show that the equation in (d) has the solution N(r)= arb 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 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 corresponding values of r and then renormalizing one map into the other.

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

The functional equation for g(x) is

g(x) = αg2(xα)

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

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