Hi everyone,

I'm trying to work through this huge problem(s) that follow. I'm studying FPI (Fixed Point Iteration) and cobweb analysis and need a lot of your help. The problem and its parts are:

Consider the logistic map $\displaystyle g : [0,1]\implies[0,1], g(x) = rx(1 - x)$, and the sequence of iterates $\displaystyle {x_0}, {x_1} = g({x_0}), ... , {x_{k+1}} = g({x_k})$. Here $\displaystyle r$ is a parameter $\displaystyle 0 < r \le 4$.

(a) How do I show that the map is well defined, that is, the range of g is included in $\displaystyle [0,1]$ for all $\displaystyle 0 < r\le 4$.

(b) How can I find the fixed points of $\displaystyle g$ as $\displaystyle 0 < r < 1 $ and determine whether the FPI method is convergent. Also use the FPI Theorem and cobweb analysis to study the evolution of the iterates $\displaystyle {x_k}$ starting with a given initial value $\displaystyle {x_0}\in{[0,1]}$.

(c) How can I find the fixed points of $\displaystyle g$ as $\displaystyle 1 < r < 3$ and determine whether the FPI method is convergent. Use the FPI Theorem and cobweb analysis to determine the evolution of the iterates $\displaystyle {x_k}$ starting with a given initial condition $\displaystyle {x_0}\in{[0,1]}$.

(d) What happens if $\displaystyle r = 1$ or $\displaystyle r = 3$?

(e) Study the local convergence of the FPI method if $\displaystyle 3 < r\le 4$.

(f) I need to use cobweb analysis to study the behavior of iterates if $\displaystyle r = 3.3$.

(g) I need to use cobweb analysis to find other values of $\displaystyle r$ for which the iterates of $\displaystyle g$ exhibit interesting behavior.

I found the wikipedia article: Logistic map - Wikipedia, the free encyclopedia

which I've been using to try to help me but I'm so lost still. I'm having the most trouble discussing what is going on at each of the questions above. Any help on this problem would be greatly appreciated!