Fixed and Periodic Points Thomas LoFaro Gustavus Adolphus College

Maple can do any algebraic or calculus computation that you can imagine. What's great is that it can do these computations exactly or approximately depending on your need. I generally use exact computations, but there are times (and we will see some here) where approximations are either necessary or more expedient. As usual, I will focus on techniques and ideas that apply to this course. However, a wide variety of example notebooks are included in Maple. To see them you can execute the following. LUklaGVscEc2IjYjUS9leGFtcGxlcy9pbmRleEYk JSFH

One of the most useful commands is solve which solves an equation (if possible). Below are 3 simple examples. In the first two Maple can solve the equation and in the third it cannot. 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

NiQsJiIiIyIiIiomI0YlRiRGJSkiIzZGJ0YlRiUsJkYkRiVGJiEiIg==

print(); 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print(); 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print(); We will often need to select from a list of solutions. This is done by putting the location of the solution we want in brackets after the variable name. Let's go back to the first example and select the second solution. 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print(); 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print(); Now the variable soln has been assigned the value of the first solution to this equation. Another useful algebraic command is factor. Let's factor our example instead of solving it. 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

LCgqJiIiJSIiIilJInhHNiIiIiNGJUYlKiYiIztGJUYnRiUhIiIiIiZGJQ==

print(); What happened? Maple assumed that since the coefficients were integers that I wanted to factor this polynomial over the integers when in fact I want it to factor over the reals. I need to include this in my factor command like this. 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

LCQqKCQiIiUiIiEiIiIsJkkieEc2IkYnJCIrW2cob1QkISM1ISIiRicsJkYpRickIismUjckZU8hIipGLkYnRic=

print(); This second argument can be a lot of things. For us the only relevant options are real and complex.

There are several ways to take derivatives. One of the easiest is to use the Expression menu on the left of the Maple window. We haven't talked about this yet, but by clicking on the operation you want you can paste the expression you need into your notebook. Thus to take the derivative of our example above we click on the derivative operator 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and replace the colored letters by the ones we want. Let's take the derivative of our example expression and evaluate it at the root we computed. 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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkldGVtcEdGJiwmKiYiIikiIiJJInhHRiZGL0YvIiM7ISIiNyNGLA==

print(); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkmbWZyYWNHRiQ2KC1GLDYlUSV0ZW1wRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkpbXBoYW50b21HRiQ2JC1GIzYlRistRiM2JUYrLUYjNiZGKy1GLDYlUSJ4RidGN0Y6LUkjbW9HRiQ2LVEiPUYnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGUS8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zqbi1GLDYlUSJhRidGN0Y6RitGKy8lLGNvbnN0cmFpbnRzR1EsaGVpZ2h0LW9ubHlGJy8lLmxpbmV0aGlja25lc3NHUSIwRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGaG8vJSliZXZlbGxlZEdGUS1GSjYtUSkmdmVyYmFyO0YnRk1GT0ZSL0ZVRjlGVkZYRlpGZm4vRmluUSYwLjBlbUYnL0Zcb0ZicC1GMjYoLUY+NiQtRiM2JUYrLUYjNiRGKy1GLDYlUSVmKHgpRidGN0Y6RitGYG8tRiM2JUYrLUYjNiVGRkZJLUYsNiVRJXNvbG5GJ0Y3RjpGK0Zjb0Zmb0Zpb0ZbcEYrRis=

SSV0ZW1wRzYi

I did both of these calculations using the Expression menu. If we define a function then we can use the ' notation to compute derivatives. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZTLUYsNiVRInhGJ0YvRjItRjY2LVEtJlJpZ2h0QXJyb3c7RidGOUY7Rj5GQEZCRkRGRkZIRlJGVEZPLUYjNistSSNtbkdGJDYkUSI0RidGOS1GNjYtUScmc2RvdDtGJ0Y5RjtGPkZARkJGREZGRkhGUkZULUklbXN1cEdGJDYlRlUtRiM2Iy1GaG42JFEiMkYnRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RKCZtaW51cztGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRmhuNiRRIzE2RidGOUZbb0ZVLUY2Ni1RJyZwbHVzO0YnRjlGO0Y+RkBGQkZERkZGSEZSRlQtRmhuNiRRIjVGJ0Y5LUYsNiNRIUYn

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkiZkdGJmYqNiNJInhHRiZGJjYkSSlvcGVyYXRvckdGJkkmYXJyb3dHRiZGJiwoKiYiIiUiIiIpRi4iIiNGNUY1KiYiIztGNUYuRjUhIiIiIiZGNUYmRiZGJjcjRiw=

print(); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RidGL0YyRjktRiw2I1EhRic=

LCYqJiIiKSIiIkkieEc2IkYlRiUiIzshIiI=

print(); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSVzb2xuRidGL0YyRjktRiw2I1EhRic=

LCQqJiIiJSIiIikiIzYjRiUiIiNGJSEiIg==

print(); JSFH

Using the tools that we learned in the previous two sections we can now use Maple to find fixed points, periodic points, and compute their stability. Before we do this let me provide a word of warning regarding periodic points. Recall that a period n point is a fixed point of 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If f is a quadratic polynomial then the degree of this expression is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYsNiVRIm5GJy8lJ2l0YWxpY0dRJXRydWVGJy9GN1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGNg== and thus if n is greater than 2 there is no guarantee that even Maple will be able to find all the periodic points (If you've had abstract algebra you should be able to tell us why this is true). Having said this, it is still worth while learning how to do this. Let's consider the example JSFH 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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkiZkdGJmYqNiNJInhHRiZGJjYkSSlvcGVyYXRvckdGJkkmYXJyb3dHRiZGJiooJCIjTSEiIiIiIkYuRjYsJkY2RjZGLkY1RjZGJkYmRiY3I0Ys

print(); and compute fixed points and period 2 points and stability. 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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkkZnBzR0YmNiQkIiIhRi4kIitITiMpZXEhIzU3JEYtRi8=

print(); There are two fixed points. Now for stability 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

JCIjTSEiIg==

print(); 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

JCErKysrKzkhIio=

print(); Because both of these are greater than 1 in absolute value, we know that both fixed points are repelling. Now for the period 2 points. 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

LV9JLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IkksbXByaW50c2xhc2hHRiY2JDcjPkkocGVyMnB0c0dGJjYmJCIiIUYuJCIrSE4jKWVxISM1JCIrJSpSYUAlKUYxJCIrd0NqPlhGMTcmRi1GL0YyRjQ=

print(); Note that we get 4 answers. This should be no great surprise since f(f(x))-x is a degree 4 polynomial. Moreover, 2 of these are really fixed points (namely the first 2). Thus there are two prime period 2 points or equivalently, 1 prime period 2 orbit. Let's check this. 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

JCIreENqPlghIzU=

print(); 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

JCIrJipSYUAlKSEjNQ==

print(); Sure enough, the last 2 numbers in the list get mapped onto each other by f. Now let's check stability. 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

JCEqKSoqKioqZighIio=

print(); We now know that this period 2 orbit is attracting. Do I need to check this at the last value in the list?

Do problems 1abce and 2abc at the end of chapter 5 in Devaney.

JSFH