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Today we're gonna talk about
chemical cycles and chaos.
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This overall idea fits into one of our
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main interests in this course which is,
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what properties and processes
are easy to obtain
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through physical dynamics alone?
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And what we'll describe here today is
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how simple systems of chemical reactions
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can lead to cycling behavior
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and can also lead to chaotic dynamics.
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So before we can go into the details
of those chemical reactions,
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and the mathematical analysis of them,
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I first want to define
a few different things.
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So the first is a limit cycle.
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What a limit cycle is,
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is imagine you have two parameters,
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X and Y, coupled together
through, let's say
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a set of differential equations.
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And a limit cycle is a case where
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the only way to define
a steady-state solution
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to this system of differential equations,
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is through a closed curve,
rather than a single point.
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So we can imagine that the steady-state
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of some system goes to
a paired value of X and Y.
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In the case of a limit cycle,
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it goes to the set of paired values X and Y
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that lie along one curve and
so will continually, over time,
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cycle through all these different values.
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Now, these limit cycles are also related
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to chaotic dynamics and we'll show how
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some of these limit cycles
become chaotic dynamics
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in our simple set of equations.
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And here is a tracing of the dynamics
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through now X, Y, and Z coordinates
projected in this 2-D space
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where you can see
that the dynamics trace out
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a large variety of combinations
of X, Y, and Z values.
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They're sort of contained in this
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ruffly understandable picture,
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but yet the state space that is being
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explored by X, Y, and Z is very large,
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and we'll talk about that today.