Chapter 10.3, Problem 11E

Interpretation Introduction

Interpretation:

Let ${\text{x}}_{\text{n+1}}\text{= f}\left({\text{x}}_{\text{n}}\right)$, where $\text{f}\left(\text{x}\right)\text{ = -}\left(\text{1 + r}\right)\text{x- }{\left(\text{x}\right)}^{\text{2}}\text{- 2}{\left(\text{x}\right)}^{\text{3}}$.

• a) To classify the linear stability of the fixed point ${\text{x}}^{*}=0$.

• b) To show that a flip bifurcation occurs at ${\text{x}}^{*}=0$ when $\text{r = 0}$.

• c) By considering the first few terms in the Taylor series for ${\text{f}}^2\text{(x)}$ or otherwise, to show that there is an unstable $\text{2-}$ cycle for $\text{r < 0}$, and this cycle coalesces with ${\text{x}}^{*}=0$ as $\text{r}\to 0$ from below.

• d) To find the long term behavior of the orbits that starts near ${\text{x}}^{*}=0$ for $\text{r < 0}$ and $\text{r > 0}$.

Concept Introduction:

• ➢ The difference equation ${\text{x}}_{\text{n+1}}\text{= f}\left({\text{x}}_{\text{n}}\right)$ where the function f is a differential function and is able to solve.

• ➢ The fixed points of the function are those points where the curve of the function meets the line.

• ➢ The equation of a period cycle $\text{-2}$ is ${\text{x}}_{\text{p}}{\text{= f}}^{\text{2}}\left({\text{x}}_{\text{p}}\right)$.