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I'd like to conclude this first unit by summarizing what we've done so far.
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The main topic of this unit is Iterated Functions.
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I begin by introducing or reminding you about functions. A function is a rule that takes a number as an input and then outputs another number.
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I've encouraged you to think about a function as an action.
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A function is something that does something to a number, and outputs a new number.
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This is a standard picture for that: x is the standard input, something happens inside this box
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- this action is called f, and it produces a new number called f of x, x after f has done its "thing".
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The functions that we will study are deterministic. That means that the output is determined only by the input,
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and if you put the same input in the function several times, you'll get the same output everytime.
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Then, we studied iterating functions. So we iterate a function by turning it into a feedback loop:
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we took a static situation, a function, and we made it dynamic by looping it around.
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So we start with some number, we apply f to it, we get a new number, then we use that number as input, apply f again and so on.
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An iterated function is an example of a dynamical system, a system that evolves in time according to a well-defined unchanging rule.
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We iterate a function by applying it again and again to a number - it's a repetitious process.
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The number we start with is known as the seed or the initial condition, and is usually denoted x-naught or x-zero.
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When you iterate a function, you get a sequence of numbers, and this sequence is called an itinerary or an orbit.
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It's also sometimes called a time series or a trajectory.
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Individual iterates are denoted by x-t, where t is the time step.
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For example, x-5 would be the fifth iterate.
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A useful way to visualize an itinerary is with a time series plot.
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So here's a time series, I plotted this itinerary and we can see that the function increases
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and then approaches what appears to be a fixed point around 4.3.
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A fixed point of a function is a number that is not changed when the function acts on it.
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For example, 1 is a fixed point of the function f of x equals x squared, because f of 1 is 1, 1 squared is 1.
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More generally, x is a fixed point if f of x equals x;
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if you input x to the function and what you get out happens to be the same thing you put in.
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There're different types of fixed points: stable and unstable.
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The fixed point is stable, if nearby points move closer to the fixed point when they're iterated.
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A stable fixed point is also called an attracting fixed point, or an attractor.
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A fixed point is unstable, if nearby points move further away from the fixed point when they're iterated.
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An unstable fixed point is also called a repelling fixed point or a repellor.
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The stability of fixed points can be summarized graphically in these pictures.
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So here's a stable or attracting fixed point - the fixed point is this point here on the line
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- and this shows that nearby points are pulled towards it. And it's an attractor.
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And here's an unstable fixed point. Here's the fix point, nearby orbits are pushed away from it;
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it's unstable or repelling. This picture can help us think about it.
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A stable fixed point is like a marble on the bottom of a bowl: if we push this to either side, it returns to its original position.
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An unstable fixed point is like a marble perched on top of a ball.
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If we push it a little bit, it moves to the right or to the left and does not return to the fixed point.
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Usually when studying dynamical systems, we 're interested in stable behavior, as this is what we're most likely to observe.
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Unstable behavior doesn't last long.
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The phase line lets us see all at once the behavior of all initial conditions.
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In a phase line, time information is not included. We can tell what direction the orbits go, but not how fast they go.
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An example, here's the phase line for the squaring function. This phase line shows that seeds larger than 1 tend towards infinity,
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1 is an unstable fixed point, seeds between 0 and 1 approach 0, and 0 is a stable fixed point.
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In dynamical systems, one of the main goals is to classify or characterize the types of behavior seen in particular classes of dynamical systems.
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So far, we've examined one class or type of dynamical system, iterated functions, and we've seen the following sorts of behaviors:
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fixed points, both stable and unstable - we've seen that orbits can approach a fixed point, get pulled towards an attractor,
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orbits can keep on growing, they can tend towards infinity, get larger and larger
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and they can also tend towards negative infinity, they keep getting more and more negative, moving more and more left on the number line.
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But this isn't the end of the story. We'll encounter more types of behavior soon.
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So at this point, I hope that iterated functions seemed - if not yet simple - then certainly simple-minded and definitely repetitious.
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All iterated functions involve is choosing a number and then applying a function to that number
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over and over and over and...over again, and see what happens.
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It's a pretty simple process really, but we will see in the weeks ahead that this simple repetitious process
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is capable of producing some surprising and complex results.
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So this brings Unit 1 to an end. A reminder that I encourage you to practice these ideas and techniques on the quizes and in the homework.
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If you have any questions about the quizes or homework, or any other course material,
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please post your questions in the forum on the course website.
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See you next week in Unit 2.