We can work on Lab exploration 2: Fractal systems

You will need install NetLogo to conduct an experiments.And using NetLogo 6.1.1 Download: http://ccl.northwestern.edu/netlogo/
Use the file TentMap and Fractals for conducting the experiment.
Lab exploration 2: Fractal systems
Math 309 Fall 2020
Deadline: 28 September
• Conduct experiments as directed.
• Journal entry. Respond to each of the “journal queries.” Using concise and clear sentences,
incorporate data, symbols, and illustrations into your text. Have an audience in mind. Focus
on developing an explanation or argument that stems from your simulations.
Submit 300-400 words 2-3 pages double-spaced to the Beachboard dropbox.
• Recommended. Work in groups of 2 or 3. Submit one journal entry for the group.
• Net Logo models are available either in the Models Library or on the course website.
Model: Tent map. Set the value of R equal to 3. Also, adjust the graph scaling parameter:
ss = .67.
1.1 Journal query.
Select an arbitrary starting point and generate its orbit. Does the orbit remain inside the interval
[0, 1]. Find a point whose orbit “escapes” from the interval. How many iterations does it take to
escape? What happens to the orbit after it’s escaped?
1.2 Journal query.
Now find a point whose orbit takes twice as many iterations to escape as the point that you
previously found. Can you continue to find points with escaping orbits that require more/many
iterations? Where’s a good place to look for such points? (Take a look at the next journal query.)
1.3 Journal query.
Are there points whose orbits don’t escape the interval? Obviously, yes. Take x0 = 0. Find five
more points whose orbits remain in [0, 1]. How many such points do you think there are? Try to
describe them in a systematic way.
1.4 Journal query.
Make a sketch of the interval from 0 to 1 that indicates which points leave the interval after one
application of the map. Now indicate which points leave after two iterations. (Coloring these sets
could help.) Three iterations? Describe the whole collection of “escaping points.” Is it a familiar
Model: Fractals. Run the Koch Curve construction for ten steps. Note the length of the “curve”
as well as the fractal dimension.
1.5 Journal query.
Next, iterate the Dragon Curve for ten steps and note the dimension and length. Briefly describe
the rule that’s followed in order to generate the figure. Briefly explain why the fractal dimension
of the dragon curve is a whole number, namely, 2.
1.6 Journal query.
Compare the lengths of the Koch and dragon curves for a given number of iterations. Does the
rate at which the length grows affect the dimension? Briefly explain.

Sample Solution

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