An Introduction to Maple for Discrete Dynamical Systems Thomas LoFaro Gustavus Adolphus College

JSFH Maple is a computer algebra system that performs an incredible variety of tasks and calculations. Our goal is to use some of these tools to help us understand discrete dynamical systems. JSFH The most important thing to realize is that help is always available. The Help menu in the toolbar provides access to tutorials and online manuals. In addition, there is a Quick Reference page that you might want to keep open (or print out) for easy reference. Quick Reference is available under the Help menu or by the (Ctrl+F2) keystroke combination. Quick Help is available by hitting F1.

In Maple you have the ability to both edit text and do mathematical computations and you need to tell the software which you are doing. This is what the Text and Math buttons on the upper left are for. To enter text click on the Text button. To enter Maple's mathematical formulas click on the Math button. (What key toggles between these two states? Go to Quick Help (F1) to find out.) The default input state is Math Mode and if you are not adding text to your worksheet you don't have to toggle between the two states. As you can tell from this document, you can add titles, sections and even change the color of text. Let's not worry about any of this. If you need it I'm sure you can figure it out. Math can be entered on a new line like this LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEjNjVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUScmcGx1cztGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkctRiw2JFEjMjRGJ0Yv

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print();

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or can even be entered in a line like this LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEjMzRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUScmc2RvdDtGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkctRiw2JFEjMjdGJ0Yv

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print();

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As you saw in the last section, Maple can be used like a calculator. The basic operands are +,-,*,/, and ^ (the last for exponents). Do some basic calculations below. You should both modify my examples calculation and insert some new ones of your own (to do this use Insert->Paragraph->...). 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print();

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print();

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Note that Maple makes the mathematics look pretty. Maple calls this formatting 2D Input. This is great, but it does take some getting used to when editing as moving around to change things can be cumbersome. The arrow keys move you around in 2D Input but not always the way you think it will. Also note that Maple does exact arithmetic (no decimal approximations). We can get it to do floating point (approximate) arithmetic most easily by using decimal points in our calculation as below. JSFH 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print();

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Note that I just needed a decimal on one of the terms. One of the most fundamental concepts that we need to understand is the difference between a set equals (:=) and a logical equals (=). A set equal (:=) is used to assign a value or expression to a variable name. Below I assign the variable foo the value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUkmbXNxcnRHRiQ2Iy1GIzYlRistSSNtbkdGJDYkUSMzN0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YrRitGKw==

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print(); JSFH 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print();

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print();

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The logical equals is used in equations and we will need this less frequently in this course. Below are a few examples so that you can see how it is used. JSFH 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print();

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print();

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JSFH

A discrete dynamical system is all about the iteration of functions. Thus it is essential that we learn how to define and manipulate functions using Maple. The syntax for a function of one variable is funcname := var -> expr. For example, to define f(x) = x^2-3 we do the following PkkiZkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmKiQ5JCIiIyIiIiMhIiJGLkYvRiRGJEYk

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print(); # input placeholder

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Note that the arrow is entered by typing minus then greater than (->); Maple then makes it look pretty. We can then use the function in the obvious ways. LUkiZkc2IjYjIiIj

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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This last calculation reminds me of a couple of relatively important things. First, the percent sign (%) is reserved to mean "the output of the preceding calculation. Secondly, the built-in function evalf() converts from exact arithmetic to floating point. Thus to convert the output from the previous line to decimals we do the following LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI=

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print(); # input placeholder

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JSFH

Now that we have defined a function, we can iterate it very easily. Let's use the function I just defined with the inital condition of 0. The basic process is as follows. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1JI21uR0YkNiRRIzAuRidGPkY+Ris=

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print();

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print();

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print();

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etc. As you can see already, this is convenient but a bit combersome. This is where the seq function comes in handy. seq is short for sequence and generates a sequence for us without having to do each of the calculations manually. For the iteration of functions the syntax is seq((f@@i)(x0), i=imin..imax). The i=imin..imax term should be read as "i goes from imin to imax in steps of 1." Thus to compute the first eleven iterates (including the initial condition) of the function above we evaluate 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

Ni0kIiIhRiQkISsrKysrXSEjNSQhKysrKytERickISsrKyt2VkYnJCErK3YkZjMkRickISt2KilwWlNGJyQhKyxMaGhMRickISssYyYqcFFGJyQhK2xWTS1ORickISsmUmVMeCRGJyQhK1VtPHdORic=

print();

Ni0kIiIhRiQkISsrKysrXSEjNSQhKysrKytERickISsrKyt2VkYnJCErK3YkZjMkRickISt2KilwWlNGJyQhKyxMaGhMRickISssYyYqcFFGJyQhK2xWTS1ORickISsmUmVMeCRGJyQhK1VtPHdORic=

JSFH Do this using exact arithmetic. Note that the @@ stands for repeated composition. Thus (f@@2)(x) = f(f(x)). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JUYrLUYjNiVGMUY6LUZVNiQtRiM2Iy1GLDYlUSJ4RidGNEY3Rj5GK0Y+Ris=

LCYqJCksJiokKUkieEc2IiIiIyIiIkYrI0YrRiohIiJGKkYrRitGLEYt

print();

LCYqJCksJiokKUkieEc2IiIiIyIiIkYrI0YrRiohIiJGKkYrRitGLEYt

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiQEYnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZURj0tSSNtbkdGJDYkUSIyRidGQUZBRistRjA2JC1GIzYjLUYsNiVRInhGJ0Y3RjpGQUYr

LCYqJCksJiokKUkieEc2IiIiIyIiIkYrI0YrRiohIiJGKkYrRitGLEYt

print();

LCYqJCksJiokKUkieEc2IiIiIyIiIkYrI0YrRiohIiJGKkYrRitGLEYt

JSFH

For the function F(x) = x^2 - 2 complete the following exercises in Maple. Define F(x) and find all values of x satisfying F(x)=x. At each of these values compute the derivative. Choose ten different initial seeds in the interval (-2,2). For each chosen seed, compute the first 100 points on the corresponding orbit. Record the results by listing the initial seed together with what happened to the orbit; that is, determine whether the orbit is fixed, periodic, eventually periodic, or has no visible pattern.

For the function G(x) = x^2 - c (you choose c) complete the following exercises in Maple. Define G(x) and find all values of x satisfying G(x)=x. At each of these values compute the derivative. Choose ten different initial seeds. For each chosen seed, compute the first 100 points on the corresponding orbit. Record the results by listing the initial seed together with what happened to the orbit; that is, determine whether the orbit is fixed, periodic, eventually periodic, or has no visible pattern.

This exercise is different from the other two so read these instructions carefully. The doubling map is defined in section 3.5 of your text. The function d(x) simply multiplies x by 2 and drops the integer part if the result is greater than 1. A piecewise definition is given in your text. Go to a spreadsheet such as Excel. The doubling function can be defined in a spreasheet by using the if function and iteration can be done using a structure like the one shown below. JSFH A B C 1 n x(n) x(n+1) 2 0 init seed =if(B2<.5,2*B2,2*B2-1) 3 1 =C2 =if(B3<.5,2*B3,2*B3-1) After entering these cells you can fill column A (right-click and drag down, choose series from drop-down menu) and simply copy cells B3 and C3 to make 100 entries. In this spreadsheet choose ten different initial seeds. Each seed should be in the interval [0,1). For each chosen seed, compute the first 100 points on the corresponding orbit. Record the results by listing the initial seed together with what happened to the orbit; that is, determine whether the orbit is fixed, periodic, eventually periodic, or has no visible pattern. Now define d(x) in Maple. Again choose ten different initial seeds (they can be the same or different as those in part 1). Each seed should be in the interval [0,1). For each chosen seed, compute the first 100 points on the corresponding orbit. Record the results by listing the initial seed together with what happened to the orbit; that is, determine whether the orbit is fixed, periodic, eventually periodic, or has no visible pattern. Read and answer the Notes and Questions on page 26 of your text.

JSFH