Math 145, Winter 99: Chaos TheoryLab #6, Instructions from Ely Kerman Introduction: Please read carefullyThis week we will be investigating the ``orbit diagram'' for the logistic equation. To quote Steven Strogatz, this is ``... a magnificent picture that has become an icon of nonlinear dynamics.'' Indeed, we will see in this example many of the special features that characterize chaotic behaviour. But first we must figure out what we will be looking at.Recall from last week that the logistic equation is given by xn+1 = rxn(1-xn).This is a dynamical scheme with the parameter r. We saw last week that for different values of r the logistic equation produced different types of sequences. For example, when r=2.8 all sequences quickly converged to 0.616. When r=3.3 the sequences eventually entered a period-2 cycle and began to repeat themselves with every other term, i.e xn=.479, xn+1 = .824, xn+2 = .479,..., etc..For r=3.5 we saw a period-4 cycle and for r=3.9 we saw no pattern at all and found numerical evidence for sensitive dependence on initial conditions. The orbit diagram will provide us with a way to picture all the different types of behaviour that occur for all relevant values of r. The values of r will be on the horizontal axis. Above each value of r we will plot values of x that characterize the behaviour of the sequences that occur for our chosen r. For example, above r=2.8 we will just plot the point x=.616 because this is the number that all sequences at r=2.8 converge to. For similar reasons, above the value r=3.3 we will plot x=.479 and x=.824. When the behaviour is chaotic (as for r=3.9) we will see a dark line above r denoting the fact that the sequences never settle down to regular behaviour. When our orbit diagram is complete we will not only be able to tell what kind of behaviour to expect for each r, we will gain a new perspective on the structure of the dynamical scheme as a whole. Exercise 1In this exercise we will use MAPLE to see what will appear over various values of r in our orbit diagram. To get the values of x that characterize the behaviour of sequences at a specific value of r we will choose an arbitrary initial condition and follow the sequence for a large number of iterations, i.e. we will calculate x_1,...,x_300. Then we will plot the last 250 values. For instance, if the sequence converges to a period-2 cycle these last 250 values should be alternating back and forth between two numbers and we would see only these two points above r. Enter the worksheet in Fig.1 and see what will appear over a variety of values of r. In particular, try the values r=2.8, 3.3, 3.5, 3.5441, 3.5645, 3.82, 3.83, 385, and 3.9. Print out the graphs for r=3.82 and r=3.83 and explain what the sequences are doing.Exercise 2. The whole enchiladaNow we are ready to look at, and interpret, the entire orbit diagram. To do this we will be using Ralph's TcI script because it is much better (and faster!!) than what I could do using MAPLE. Get to the class webpage and goto ``Suggestions from Ralph'' in Lab 6 (Week 7). Save the file ``Version02, with sliders'' to the desktop and then exit netscape. Now open the file named Wish (I had to use the find function to do this). Goto ``source'' under the File menu and locate the file we just downloaded. Double-click on this and you should see the whole orbit diagram.Notice that mixed into the broad ranges of chaotic behaviour we have smaller ranges of orderly behavior. These are called ``periodic windows'' and largest of these starts near r=3.83. Using the sliders set $cmin=3.847 and $cmax=3.857. Amazingly we can see small copies of the whole diagram at smaller scales. This phenomenon is known as ``self-similarity'' or ``universality'' and it reflects the fact that no matter how much you zoom in on part the picture the structure remains just as complex. Try to fucus in on other periodic windows or any other part of the diagram which interests you. Your Portfolio: Fact Sheet and ChecklistFacts
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