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Hi everyone. Welcome to unit 2 of the course.
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In this unit I'll introduce another type of dynamical system: differential equations
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but before I do I want to say a bit more about iterated functions.
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I'll quickly review some of the ideas from last unit,
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and more importantly, I'll underscore some concepts in dynamical systems
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that will be useful as we start thinking about differential equations.
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So, let's get started.
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So, let's return again to iterated functions,
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and I want to use iterated functions to introduce a little bit of new terminology and language
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that will be useful when we look at differential equations.
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So, an iterated function is a function that's turned into a feedback loop.
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So, for a concrete example: suppose that the function is 1/2 X + 3
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and I choose a seed of 2
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Then, what's the first iterate? Well, I need to apply the function to the number
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So, 1/2 of 2 is 1 , 1 + 3 is 4
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What's the next value? X2, the second iterate. The function tells me.
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I apply the function to 4
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A 1/2 of 4 is 2 , 2 + 3 is 5
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and I can keep going, you're familiar with this process now,
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and I can form the itinerary.
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So the function determines the itinerary.
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So let me write this in a slightly different way.
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So, in this notation, this tells me that the next X, the next value in the orbit
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is equal to the current value Xn after the function is applied to it.
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So, I can think of this equation, which really says in symbols what this picture up here says in picture,
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that the function gives you the next iterate, or orbit,
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You can think of the function as a rule applied again and again that specifies the orbit or the itinerary.