Chapter 10.7, Problem 10E

Interpretation Introduction

Interpretation:

To fill in the missing algebraic steps in the concrete renormalization calculation of period doubling and to verify that the following equations are correct:

${\text{η}}_{\text{n+1}}{\text{= (1- 4μ - μ}}^{\text{2}}{\text{)η}}_{\text{n}}{\text{+ Cη}}_{\text{n}}^{\text{2}}\text{+}\mathrm{.....}$ Where, ${\text{C = 4μ + μ}}^{\text{2}}\text{- 3}\sqrt{{\text{μ}}^{\text{2}}\text{+ 4μ}}$

Concept Introduction:

• Renormalization is the self-similarity of the Figtree. The twigs look like the earlier branches, but they are scaled down in both the x and ther directions. This structure reflects the endless repetition of the same dynamical processes.

• A period doubling bifurcation is a bifurcation in which a slight change in the parameter of the system equationsgenerate the system modifications to a new behaviour with two fold the period of the original system.