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So, I think the phenomenon of universality
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in physical systems is incredibly amazing.
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We see period doubling in physical systems
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and that period doubling is described by a
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delta, and that delta has a value that is
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the same as for these one-dimensional
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systems that we started studying with -
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the logistic equation, and so on.
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If you're not already convinced that this
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is amazing, let me try to convince you by
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going over the line of reasoning that led
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us to this point. So, I started a long
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time ago in this course talking about the
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logistic equation. And when I presented it
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I said don't take this equation very seriously
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- it is certainly not a realistic model of
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how rabbits might grow, or how almost anything
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might grow. But it's more like a characature
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or sketch or cartoon - it gives us some
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rough picture of how a population might grow
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where there is some limit to the growth.
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And then we found that there is surprising
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richness to this equation, amazing richness
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when we form the bifurcation diagram and
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zoomed in again and again and again and saw
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these fractal patterns and complicated behavior.
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But we might have thought...it would be very
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reasonable to think "well we're just playing
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with a mathematical game - it's just an
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equation that somebody told a story about
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rabbits to motivate - but it's just a little
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equation and it's really neat that it does
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all this stuff - but what might it say about
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the real world?" And we talked about that sum,
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and I appealed to the argument that this
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shows that even the simplest possible...
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or just about the simplest possible equation
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...iterated function...can show the butterfly effect
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and sensitive dependence and that's an
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important realization and that certainly is.
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But still the logistic equation seemed just
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like a little toy model - a fun thing to play with.
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Then, in this chapter, we started looking at
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...or in this unit...we started looking at
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bifurcation diagrams for different equations
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- the sine equation, the cubic equation, we
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could have done lots of others - and we saw
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that the bifurcation diagram was kind of
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similar. Then, I presented the notion of this
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quantity delta, looking at the ratios of the lengths
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of period two and then period four and then
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period eight regions. And we saw that
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the same number appeared - 4.669201 - so
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that all of these different functions turn out
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to behave the same...not exactly the same,
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but they undergo this transition to chaos
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from periodic to chaotic behavior in the same way.
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We're still in the realm of math. It's an amazing
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mathematical fact that we're still in the
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realm of math. We might wonder what this has
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to say about the real world. So now,
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experimentalists, scientists, and physicists have
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gone out and actually measured period-doubling
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in a whole bunch of different physical phenomena
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and they've calculated this quantity delta -
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this ratio of period one to period two,
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period two to period four, and so on. And they
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find numbers consistent with that predicted by
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this theory - 4.669. So, we started with a
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ridiculously simple, deliberately unrealistic
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equation, and nevertheless, that equation
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contains some information in the form of a
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number that one can go out and measure in real
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physical systems. So this surely tells us something
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important and profound - that nature and mathematics
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are somehow constrained in the ways that a transition
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from periodic to aperiodic behavior - from
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periodic-ordered behavior to chaotic
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behavior - can occur. That when it undergoes
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period-doubling, almost always what it's going
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to do in the large-end limit, it's going to do this
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transition in the same way. So all of these different
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phenomena - period-doubling in rabbit equations
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on the computer and in sine equations on the computer
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, and in electronic circuits, and in convection rolls,
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and more - all have some similar feature about them.
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There is some way that this transition is constrained
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and that certain featyres can only occur in one way. So this also
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says that some of the details of our mathematical models
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apparently don't matter because we see the
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same behavior across a very wide range of systems.
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That's what it means to say that this quantity
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delta is universal. So, of course this leads to
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a bunch of questions - how can this possibly be?
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And there are at least two parts of that question
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I think. The first is a mathematical one -
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how can it be that all these different one-dimensional
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functions have the same delta? That they undergo
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period-doubling in essentially the same way -
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this 4.669 number. The other question is
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what in the world would a dripping faucet have
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to do with a one-dimensional equation? Or fluid
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mechanics where we have convection rolls wiggling
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around in a little box? These are surely high-dimensional
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systems - you would model them with more than just
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a one-dimensional equation, and they're also
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continuous systems - they're not discrete like
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iterated functions. The second question - what
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one-dimensional iterated functions might have to do
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with higher-dimensional real physical systems - I'll
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discuss in the next several units. In the next sub-unit ,
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the next set of lectures here, I'll try to give some feel
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for how it can be mathematically that there are
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these similarities among all of these different
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one-dimensional functions.