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