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In this unit, I will talk about fixed points
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and stability.
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Two important concepts we will use
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throughout the course.
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But first, I should mention
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that iterated functions are an example
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of a dynamical system
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and the course is about dynamical systems.
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So, this is important.
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So I should say what dynamical systems are.
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And I'll come back to this again.
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But, for now:
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Dynamical system is a system
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that evolves in forward forward in time
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according to a well defined and unchanging rule.
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And that is what we have for iterated functions.
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So, we have a value, a number, that moves forward
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changes from iterate to iterate.
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According to a well defined rule.
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That is the function. A nice deterministic function.
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The same input gives the same output each time.
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And the rule doesn't change as we iterate.
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We are doing the same thing over and over again.
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Using the output of one step
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for the input of the next
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So, iterated functions are a dynamical system
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In the study of dynamical systems,
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We are often not particularly interested in the numbers
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of a particular orbit or itinerary.
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Instead we are interested collections of orbits,
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many orbits all at once.
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And more generally we would like to understand
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What sort of behaviors do we see in different types
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of dynamical systems?
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So this is the approach we will take in this course
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and so we will start that in this unit
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When we look at fixed points
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and stability
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Let's start with an example.
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We will consider the squaring function.
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F(x) = x^2.
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We know how to calculate an orbit for this function.
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We just start with the number, the seed.
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In this case I choose 1.1.
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And then we square it to get the next iterate.
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1.21.
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We square that to get 1.46.
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And so on.
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We could choose a different seed.
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And we would get a different orbit.
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Let's say instead of 1.1, I choose 1.2.
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I would do the same thing
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Square again and again.
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To get that orbit.
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So here is the orbit for the seed 1.2.
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Note that both orbits get larger
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when you square a number larger than 1.
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The number gets larger.
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So these numbers will continue to grow
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Let's try another one.
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Suppose I choose a seed of 0.9.
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I'll put 0.9 here.
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What will happen?
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Well let's apply the function and see.
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For this initial condition,
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the number gets smaller.
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It's getting closer and closer to zero.
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If you square a number between 0 and 1,
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It gets smaller not larger.
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Let's try one more initial condition.
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One more seed.
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I'll try 0.8.
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And I will square again and again to get the orbit.
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So again when we square a number between 0 and 1,
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it gets smaller.
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Here we can see the number gets close and closer to zero.
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So, calculating the orbit
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a particular orbit for a particular seed is not too difficult.
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It just requires a little bit of calculator work.
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But it doesn't let us get a sense of the big picture.
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What does this function do?
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So we will use some graphical techniques
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that will help us see this better
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and understand the function all at once.
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First, let me plot the time series plots
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for these four orbits.
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1.1 and 1.2 they get bigger.
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0.9 and 0.8 they get smaller.
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So here is the time series plot for that.
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You can see I have four different initial conditions.
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A square, a diamond --sorry
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-- a sqaure, a triangle, a circle, and a diamond.
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The square that is 1.2.
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We can see that growing quite fast.
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It goes off the graph.
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The triangles are 1.1
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Those grow.
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The circles are 0.9 and those are getting close to zero.
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And then 0.8 also gets close to zero.
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So this lets us see
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the numbers larger than 1 will get larger and larger
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And numbers between 0 and 1
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we suspect will get closer and closer to zero.