Investigating Iterative Maps in Maple and an introduction to Maple procedures LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic= Simple example of an iterative map, considered on [0,1] QyQ+SSJnRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQ5JCIiI0YlRiVGJSIiIg==

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Find the fixed points analytically (trivial, just to remind you of the syntax) QyQtSSZzb2x2ZUc2IjYjLy1JImdHRiU2I0kieEdGJUYrIiIi

NiQiIiEiIiI=

Numerically compute the iteration sequence starting at 3/4 QyQ+JkkiWEc2IjYjIiIhIyIiJCIiJSIiIg==

IyIiJCIiJQ==

PyhJImlHNiIiIiJGJSIiJkkldHJ1ZUclKnByb3RlY3RlZEc+JkkiWEdGJDYjRiMtSSZldmFsZkdGKDYjLUkiZ0dGJDYjJkYrNiMsJkYjRiUhIiJGJQ==

JCIrKysrRGMhIzU=

JCIrK0Qxa0ohIzU=

JCIrXSJINisiISM1

JCIrdiZmQSsiISM2

JCIrY1VfLzUhIzg=

Let's try to make the cobweb diagram of this sequence. It's a bit tricky, we'll go through step by step. It is generated by drawing a sequence of line segments from the points [x0 x0] to [x0, x1=g(x0)] then to [x1,x1] to [x1, x2=g(x2)] etc. From the point x we could generate two points in the sequence above using the command below. Note that this is how to specify a function that returns two vector quantities with two components each QyQ+SSZzdGFpckc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJTYkNyQ5JEYuNyRGLi1JImdHRiU2I0YuRiVGJUYlIiIi

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNiQ3JDkkRis3JEYrLUkiZ0dGJTYjRitGJUYlRiU=

Test it out. QyQtSSZzdGFpckc2IjYjJkkiWEdGJTYjIiIhIiIi

NiQ3JCMiIiQiIiVGJDckRiQjIiIqIiM7

We can make the sequence of line segment points for the iteration above using the command QyQ+SSdjb2J3ZWJHNiI3Iy1JJHNlcUclKnByb3RlY3RlZEc2JC1JJnN0YWlyR0YlNiMmSSJYR0YlNiNJImlHRiUvRjE7IiIhIiImIiIi

Ny43JCMiIiQiIiVGJDckRiQjIiIqIiM7NyQkIisrKytEYyEjNUYsNyRGLCQiKytEMWtKRi43JEYwRjA3JEYwJCIrXSJINisiRi43JEY0RjQ3JEY0JCIrdiZmQSsiISM2NyRGOEY4NyRGOCQiK2NVXy81ISM4NyRGPUY9NyRGPSQiKyIpKm8hNDUhIzw=

Now plot the cobweb and display it with the curves from the beginning. QyQ+SSpTdGFpcmNhc2VHNiItSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiRJJ2NvYndlYkdGJS9JJmNvbG9yR0YlSSZibGFja0dGJSIiIg==

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Now, this is a procedure that you might want to run several times with different g, x0, and n plotting on different intervals [a,b] with different intervals [ymin, ymax] displayed in y, to avoid seeing sequences that move far away. For example, this might help with the assignment due Friday. We can make Maple lines of code reusable by putting it into a procedure, "proc" As with for loops, I can make a new line to help formatting using shift enter The local command declares variables that are only used and accessible within the procedure The uses command declares packages that will be used by the procedure 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Now let's try it out, looking back to get the order of the inputs right LUkqc3RhaXJjYXNlRzYiNilmKjYjSSJ4R0YkRiQ2JEkpb3BlcmF0b3JHRiRJJmFycm93R0YkRiQqJDkkIiIjRiRGJEYkIyIiJCIiJUYwIiIhIiIiRjJGMw==

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