1) Before you begin to use the computer, make sure you can explain the basic difference between stable and sensitive dynamical systems. This lab will continue the work you did in class with the graphing calculators. If you have not finished that previous work, sit at one of the big tables and do it now.
2) Log on to your account, choose the icon for "Dynamical Systems" and wait for it to load. From the main menu, pick "Stability versus Sensitivity." The program has been set up to run two sets of data simultaneously, using seeds which differ by only a small amount. That amount of difference is called the delta of the experiment. For example, if you set delta at .01, then your seeds will be 234.01 and 234.02. You will be able to set the delta at any desired level, within reason. The computer can handle up to 7 zeros, I believe.
3) Start with parameter a set at 3.7 and delta at .01. The data will appear on the screen in blocks of 20 iterations. Just hit Enter to see the next block. If you wish to reset the experiment, hit "d" to change delta or "a" to change parameter a.
4) You learned in class that systems with a = 3.3 are stable, while systems with a = 3.7 are sensitive. But where is the dividing line between stability and sensitivity? Start looking for that by experimenting with a variety of values for parameter a.
5) The computer will also check to see when the two Pn values are very close to each other. The computer will also check to see when the two Pn values are very close to each other. When they are within .01 of each other, the numbers turn purple. Then when they are even closer, they turn blue. Closer still, within .0000001, they turn yellow. Sometimes you have to wait quite a while before they get close, so don't give up too quickly. The other thing that might happen is that both sequences fall into a cycle, but the cycles are out of phase with each other. Call me over and I will explain more about that.
6) Now set the value of parameter a at 3.83. Run the same sensitivity experiments. Is this system stable or sensitive?
7) Even though 3.83 is a higher value of parameter a than 3.7, it turns out to be a nice stable three-cycle. This is one example of an island of stability occuring in a sea of chaos. How big is that island? How much can you increase the value of a past 3.83 and still get a stable system?
8) Look for other islands of stability. You can use the bifurcation diagram to do that. Hit Esc to get out of your experiment, and then choose Bifurcation Diagram from the main menu. Set the minimum a value at 3.5 and the maximum a value at 4. Get an idea of which regions look like they have nice, calm systems that go to a stable cycle. Then try out the sensitivity experiments on those a values. See how many different cycles you can find for a values greater than 3.57.
9) You can also use the "Check for Cycles" feature. I programmed this to remember a number and then scan subsequent iterations looking to see when that particular number occurs again. The precision level determines how much the numbers agree with each other. For instance, if you use precision of 1, you will see all values that agree with the integer portion of the remembered number. With precision of 10, they will agree up to the tenths place. The higher the precision you request, the fewer values you will see on the screen. When the computer finds an exact match (up to the trillionths place), the sequence will stop and you will be able to see the period of the cycle by looking at the difference in the iteration numbers. Sometimes this can take a long time to occur. If you decide you want to bail, just hit a key while the numbers are running and the sequence will stop and allow you to reset the experiment.
10) When you find one of the cycles, check out what its web diagram looks like.
11) Call me over to explain what "Higher Iterates" is about. Then I will give you a task to do using that technique.
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