Logistic Map and cobweb diagrams

[S.Parker]

Hi all. I am new here and I am having difficulty figuring out what exactly is required of me in this question. If someone could be so kind as to explain. For this part of the project we will consider the evolution of a discrete dynamical system given by a logistic map.
We will consider a logistic map given by

xn+1 = 2xn(1- xn) (1)

on the interval x 2 [0; 1]. A fixed point of this equation is obtained when xn+1 = xn, i.e. when

x = 2x(1 - x):

If we let

g(x) = 2x(1 - x)

then a fixed point is obtained when

x = g(x):

In our case, we can solve for this analytically, but we will investigate how a numerical solution will converge on
the solution. What we want to know is
Does the system evolve to a stable solution? i.e. is the fixed point stable?
If so, what value of x does the system evolve to?
To see how this works, essentially we will find the intersection of the straight line y = x with the function
g(x). Start by plotting y = x and g(x) on the same se of axes. Now let us start with an initial value of x and
see how the system evolves from there. Choose a suitable value of N for the number of iterations. Choose
2
x0 = 0:01, and then calculate each new value of xn from this starting value, using (1). Do this by creating a vector
x = [x0; x1; x2; : : : xN]. Plot a graph of x vs step number n. This will show you how x evolves with each new

Homework Equations

The Attempt at a Solution

To be honest, I have not tried it yet, as I am not sure of what to do exactly. I know this part is required, but I really do need the help (project due this Friday)

 

[jackmell]

I am not sure of what to do exactly. I know this part is required, but I really do need the help (project due this Friday)

I believe the best approach is for you to find "Chaos" by Peitigen and go directly to the chapter on Deterministic Chaos. Never heard it called cob webs but you'll see what they're doing with the logistic map which I guess looks like webs.