We've been talking about how
chemical dynamics can naturally lead
to cycling behavior and chaos,
and here we want to make that a little bit more explicit.
A classic model that's been used
to look at these types of phenomenas
is known as the Brusselator.
It's described by this abstract set
of chemical reactions listed here
in the middle of the screen.
Now, these chemical reactions
can be converted into a set of two differential equations
assuming that we hold the concentrations
of A and B constant in the system.
And in this case, we've added
these different reaction rates K1 and K2
which describe the reaction rates
associated with the arrows above.
The system can be easily non-dimensionalized
to a slightly simpler system where we've combined parameters
into effective parameters little a, little b, and so forth.
This system has a simple steady state at
x = a (this effective parameter), and
y = b / a (the ratio of effective parameters).
Furthermore, what's very interesting about this system is that,
we're intersted in not only what the steady state is,
but in the dynamics leading to that steady state
and whether the steady state is stable.
Using techniques known as
stability analysis which is an entire field
it's possible to show that this steady state
will be unstable for b > a^2 + 1.
So let's take a look at what that looks like.
If b is safely smaller than a^2 + 1,
then you can start off at some initial
concentrations for x and y
and these two chemical species will have
some transient behavior that eventually leads to
these two steady states that we described
in the previous slides.
Now, if we make b just smaller than a^2 + 1,
so we make it very close to a^2 + 1,
we see that you go the same steady states
but now, in the transient behavior
we pick up these oscillations that
over time, ring down,
and it takes us a long time
to go to the actual steady state itself.
And if we make b just greater than a^2 + 1
so this is the case where things should be unstable,
we now see that we have oscillations that go on forever.
So this system naturally goes through an oscillation;
naturally returns to specific states periodically,
and in this case, how fast that's occuring
how frequently these states are passed through,
that rate is increasing in time.
Now another way that we can look
at these steady states in the set of dynamics,
is in a phase diagram of x against y.
So this is looking at the paired values
of these two concetrations, x and y.
So time isn't explicit except in animation
and in this first case where b
is much less, or significantly less
than a^2 + 1, then we'll see that the
dynamics have the simple case where you
start off from any point in the space
and you simply go to this one steady state.
There's a little bit of transient behavior
but we go to a single point fixed system.
This case where b is just less than a^2 +1
where we had this ring down behavior to a
steady state in oscillations to the transient,
will look something like this in the base plot.
So you can see that we spiral into
the steady state that we expect to find.
So there's a very interesting
oscillatory transient dynamics.
And if we go above a^2 +1
where we expect the system to be unstable,
then we get this limit cycle.
So we are constantly gonna pass through
this set of paired x and y values
periodically with an increasing rate.
What's interesting about these dynamics
is that they're fully deterministic.
So we haven't put any stochasticity in the system.
We have this completely deterministic way
of getting oscillations in time, and so
it says that just the inherent dynamics
of chemical dynamics, even without noise,
can lead to really interesting oscillatory behavior.
And that oscillatory behavior can be
even more complicated than what we've been describing.
So a classic system for looking at chaos
is the Lorenz System.
And so now, we have a coupled value,
or we have a triplet of values of x, y, and z concentrations,
each related to these differential equations
that I'm showing here in non-dimensional form.
This was originally discovered
in atmospheric dynamics by Ed Lorenz
and can be mapped onto
a variety of systems including certain chemical dynamics.
Now what's interesting about this case
is that this system gives rise also to fixed steady states,
so where x, y, and z will go to a single value,
but for other parameter values in the system,
we'll find that you go to chaotic dynamics.
And here is what those chaotic dynamics looks like.
So we're only looking at x, y.
So there's a set of z values
associated with all of these,
and we're just looking at
the projection into the x, y space.
In doing that, we see that we have this
very complicated oscillation where
you not only oscillate over a set
of complicated values in x and y,
but you also transition between
these two basins of attraction.