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Hello and welcome to unit eight
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This unit is on strange attractors.
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Strange attractors are mathematical
objects
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that reside in phase space that combine
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elements of order and disorder
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predictability and unpredictability.
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In some fun and interesting ways
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I'll introduce strange attractors through
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a couple of examples. We'll look at the
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Hénon map and the Lorenz equations
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Which we study previously on unit 7
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and we'll see that they both
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posses strange attractors.
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And I'll show you what that means in
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different ways of looking at them.
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And
then I'll talk a little bit more generally
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about transformations and phase space
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and in the summary for this unit I'll give
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some remarks on why I think strange
attractors
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are interesting and fun and important.
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So let's get started by looking again at
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the Hénon map.
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So here is the Hénon map, it's an iterated
function
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a two-dimensional iterated function.
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There are two variables, X & Y
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And the next X & Y value is just a
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function of the previous X & Y values
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and this tells you what the function is
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So you start with an initial condition in
X & Y
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and then this gives you a time series
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of a X's and a time series of Y's
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So we can choose some parameter
values
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and to start [inaudible]
I choose "a" at 0.8
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and "b" 0.3 and we need to choose initial
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conditions, I think I chose 0.2 0.2
for X & Y
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and then we just apply this rule
over and over again
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obediently iterating the function as
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we've done throughout the course
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and If we did that on a program on the web
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which I'll show you, again in a second but
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just here's a screen-shot from it.
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One can see that the orbit pretty quickly
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approaches a cyclic period of two.
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So this is the time series for X
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This is the time series for Y and they're
both, wiggling, repeating every two steps
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If one look closely at the numbers
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one would find that it's oscillating
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between this two point so we have
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[inaudible] like this, an X of about -0.38
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so that is this value here
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and when X is -0.38 it turns out that Y is
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about +0.38
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And then the large X value that goes with
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the small Y value, the large X value is
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about 1.3 it turns out about 1.26 and then
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this, right? so large X, small Y.
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This value turns out to be about -0.12
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It's a plus. So the points are oscillating
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between this and this, back and forth
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between this and that. So I can draw
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a final state diagram like we did for
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the logistic equation but now the final
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state diagram is two-dimensional because
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is X and Y, instead of just X
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For periodic behavior on the logistic
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equation, the final state diagram is two points
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on a line, here the final state diagram
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will be two points on a plane. So let me
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draw that and I'll make a rough sketch of
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what this might look like. And, then we'll
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see on the computer screen tight
(marker on paper sound)
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So I do really rough sketch of this
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is -0.5; +0.5; +1; +1.5; -0.5. Ok
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So let's see we got two points, the first
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point is X -0.38 Y is +0.38
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so I go to the left -0.4 up about 0.4
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And that would be the point on the XY
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plane, then the other point +1.26 that's
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out about here, -.12 that's probably
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around here. So again the scales are not
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perfect it isn't a beautiful diagram but
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the main point is that the final state
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diagram for the Henon map when we have
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periodic behavior is just two points on a
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plane. And remember the final state
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diagram, I'll show you this again
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We're interested just in the long-term
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behavior, so I might not plot in this case
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the first ten, and then plot the remaining
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ten, and then all those points to be right
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on top of each other because their cycling
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in the same point. So we'll go to look at
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the computer screen version of this in
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just a second and we'll look at this
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This set of parameter values and then
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we'll go to the parameter values that we
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ended with. We'll make a=1.4 and we'll see
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a much more
interesting final state diagram