Assignmnent: Problem 1 from syllabus

Problem number 1) from the syllabus page of the website is stated as
1) Consider the iteration , x(n+1)=f(x(n)), where f(x) = Cx(1-x), for various different values of C. Approximately what range of C gives you chaos? Approximately what range of C does not give you chaos? Do you see any fractals? Do you see any evidence of a connection between chaos and fractals?

As far as I can see -2<4>

• C = -1:

• C=1:

• C=2:

• C=3:

I wasn't able to recognize any fractals when investigating this IFS. This may be because of the initial condition I used (x(0) = 0.3). I also can't put together the link between chaos and fractals as of this time.

Assignment: Feathered Fractal (Toy)

This fractal is similar to a Cantor set. It has no overlaps and comes from the map of the union of two transformations f and g.

This is the code I used in Mathematica:

This is the fractal, broken down between 10 iterations of the transformation:

This is the graph after 15 iterations:

Assignment (Binary String vs. IFS)

This assignment was stated as follows:

Consider the chaotic iteration x(n+1)=f(x(n)), where f(x) = 4x(1-x), and initial condition x(0)= 0.3. Is the sequence of numbers x(n) for n=0,1,2,3,4, . . . random or deterministic? Now consider the function h(x) defined by h(x)=0 for x<0.5>h(x)=1 for x>=0.5, and also consider the sequence of bits y(n)=h(x(n)), where x(n) comes from the chaotic iteration above. Is the sequence y(n) deterministic or random? Here are two sequences of bits, each 1000 steps long. One sequence comes from a chaotic iteration for a certain choice of intial condition, the other is random (or is it?). Can you tell the difference? Which is which?

As for the first part of the assignment, to find whether the sequence was random or deterministic I used Mathematica to make a plot of the system, this it what it looked like:
This leads me to believe that it was not deterministic. There doesn't exactly look as if there is much rhyme or reason to where the points are mapped to. Some spots on the graph look as if there is a pattern trying to emerge (i.e. the white spots along between .6 and .8 along the horizontal), but I'm not convinced that it shows the IFS to be deterministic.

As for the second part of this problem, we were to determine whether a mapping of this IFS onto a binary string would be deterministic or random and to find the difference between a random string and a string mapped from an IFS with a specific initial condition.

This was fairly difficult for me, but to do this I used Mathematica to help me determine which was which. First I copied our 2 given binary strings into separate lists in Mathematica.

• List a) had 510 zeros and 490 ones
  

• List b) had 495 zeros and 505 ones

This was not very helpful in finding which was the "random" string and which was produced through the IFS, since both are so close to 50% zeros and 50% ones, which would be what you might think the random string would have to be. I next created a function in Mathematica equal to the function given above and began guessing initial conditions to start with and iterating 1000 times, then mapping the list using the parameters given for h(x). Finally I subtracted this list from each of our given binary strings, and stopped once Mathematica gave a list of all zeros.

This is what the input and output looked like: {v4 being List b) from problem 2}

Thus, List a) is the coin flip, and List b) is x(n+1) = f(x(n)) with initial condition x(0) = 0.3
I can't say that I think this IFS mapped to a binary string using h(x) is deterministic. The outcome is very close to 50% of each number which is very close to the probability of a random number generator spitting out either 1 or 0.

Fractal dimension of a fractal determines its complexity. (What does this mean?)

This was a tough one. I feel like everything that has to do with fractals is very ambiguously defined. Even the definition of a fractal seems to be up in the air, defined with more, or less parameters depending on who or what your source is.

Upon first look at the statement,"the fractal dimension of an object determines its complexity" I would have to think that the fractal dimension has something to do with how intricate the rendering of the fractal is from a visual standpoint. When I think of the word complex it leads me to thinking that something looks confusing, or is dense with information. Thus, I believe the actual confusion lies on the word, complex. What does it mean?

The Oxford Dictionary defines something as "complex" if it "is made of closely connected parts". This could mean that something would be more complex if it had more closely connected parts. I believe this gives us a good realization of what the "complexity of a fractal" could mean.