Chapter 10.7, Problem 9E

Interpretation Introduction

Interpretation:

Show if the renormalization procedure in Exercise $\mathrm{10.7.8}$ led to the functional equation $\text{g}\left(\text{x}\right){\text{ = αg}}^2\left(\text{x/α}\right)$ with new boundary conditions appropriate to the tangent bifurcation: $\text{g}\left(\text{0}\right)\text{ = }0$, $\text{g'}\left(\text{0}\right)\text{ = 1}$. To verify that a solution is $\text{α=2 }$, $\text{g}\left(\text{x}\right)\text{ = }\frac{\text{x}}{\left(1+\text{ax}\right)}$, with $\text{a}$ arbitrary. To explain why $\text{α=2 }$ is almost obvious, in retrospect.

Concept Introduction:

Obtain the expression for ${\text{αg}}^{\text{2}}\left(\text{x/α}\right)$ from the given function.

Compare $\text{g}\left(\text{x}\right)$ and ${\text{αg}}^2\left(\text{x/α}\right)$.

Obtain the expression for $\text{α}$ from the $\text{g}\left(\text{x}\right)$ and ${\text{αg}}^2\left(\text{x/α}\right)$.

Plot the cobweb.