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.