Brownian Motion Part II

As I said in my last post regarding the section in which we were discussing Brownian motion, I was working on using mathematica to graph an illsutration of what Brownian motion looked like. As we discussed in class it looks very much like a Stock Market diagram (and in fact Brownian Motion is a large part of Economics and Finance). I was able to complete this task and constructed a graph of a random walk to illustrate Brownian motion in one dimension.

• One Dimensional "Drunkards Walk"

• In the example above the points are disconnected, but since they are so close together, you can still see the outline of the graph.
• In the example below I have connected the points and exectued the same piece of code. Each time it is executed, a new graph is formed due to the natural randomness of brownian motion.

• Code:
  RandomWalk[n_] := NestList[(# + (-1)^Random[Integer]) &, 0, n];
  ListPlot[RandomWalk[3000], PlotJoined -> True]

This is a cool website I came across with small applets showing both one and two dimensional brownian motion: Click Here

In class we did a sort of extended proof that Brownian motion exists (about 2.5 class sessions). This proof had, what seemed to be quite a bit of statistics involved in it and I will not attempt to rewrite the 5 pages of notes I have on it in this blog. I will however discuss what I got out of those classes:

1. Brownian motion exists.
   This was the conclusion we came to in our extended proof.
2. The paths of Brownian motion are continuous.
   This would be fairly easy to justify with the definition of continous.
3. The paths are non-differentiable anywhere.
   Holy cow! Continous, but not differentiable anwhere?
4. The paths to Brownian motion are fractals.
   Ah, yes. What I was waiting for all along, a tie in to fractals. This webpage made this part a little more clear in my eyes. It states that Brownian motion is self-similar in law, then has an applet which zooms in continuously on a random walk. You can see that the graph stretches a bit, but keeps it's generaly shape no matter how much you zoom in.
   

Correction to Bifurcation Diagram

In assignment #1 we were asked about the Iterated function system:

x(n+1) = f(x(n))
where f(x) = Cx(1-x), for some range of values for C

I had been coding this incorrectly in mathematica, I recently found out how to code it correctly, and thusly would like to make some corrections to my conclusions about this assignment.
Code:

g[x_] := a x (1 - x);
plist = {};
Do[x = 0.3; Do[x = g[x], {1000}];
Do[x = g[x]; plist = Append[plist, {a, x}], {1000}], {a, 2.8, 4, 10^-2}];
ListPlot[plist];


Approximately what range of C gives you chaos?

• Well as far as I can see the diagram looks more "chaotic" in the darker areas, and less chaotic in the white strips. Upon first look when C <~ 3.48 the graph just looks like a binary tree that keeps branching off two at a time, after that it becomes somewhat of a mess. The diagram does seem to have certain "jumps" though, where it clears up from time to time.

Approximately what range of C does not give you chaos?

• This diagram is definitely not chaotic before C = 3.48. After that it seems like it alternates between chaos and white space. If you zoom in on certain parts, you can see a simlilar pattern to the graph where C <>

Do you see any fractals?

• Yes. I see some self similarity anyway. This leads me to believe that there are fractals in this mapping.

Do you see a connection between chaos and fractals?

• There is obviously a connection between these two things, whenever there are fractals, chaos seems to be lurking around somewhere, be it in the form of randomness (i.e. Brownian motion) or the dark areas that seemingly come out of nowhere in this bifurcation mapping.