Hi all. Im new here and Im having difficulty figuring out what exactly is required of me in this question. If someone could be so kind as to explain.

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)

**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 newWe 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

**2. Relevant equations****3. 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)

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