Continuing with our review,
the next topics that we covered concern bifurcation diagrams.
They're a way to see how the behavior of a dynamical system changes as a parameter
is changed.
I think it's best to think of them as being built up one parameter value
at a time.
So, for each parameter value, make a phase line if it's a differential equation,
or a final-state diagram for an iterated function.
And you get a collection of these, and then you glue these together to make a
bifurcation diagram.
So, here's one of the first bifurcation diagrams we looked at.
This is the logistic equation with harvest. The equation is down here.
And so, h is the parameter that I'm changing.
H is here, it goes from 0 to 100 to 200 and so on.
And so, a way to interpret this is suppose you want to know what's going on
at h is 100. Well, I would try to focus right on that value, and I can see "aha,"
it looks to me like there is an attracting fixed point here, and a repelling
fixed point there. So there are two fixed points: one of them attracting and one of
them repelling, or repulsive.
And, what's interesting about this is that so this is is the stable fixed point,
this would be the stable population of the story I told involved fish in a lake
or an ocean, and h is the fishing rate, how many fish you catch every year.
And that increases, and as you increase h, the sort of steady state population of the
fish decreases, that makes sense.
But what's surprising is that when you're here, and you make a tiny increase
in the fishing rate, the steady-state population crashes and in fact disappears.
The population crashes.
So, you have a small change in h leading to a very large qualitative change in the
fish behavior.
So, this is an example of a bifurcation that occurs right here.
It's a sudden qualitative change in the system's behavior as a parameter
is varied slowly and continuously.
So, we looked at bifurcation diagrams for differential equations and we saw the
surprising discontinuous behavior, then we looked at bifurcation diagrams for the
logistic equation, and we saw bifurcations here is period 2 to period 4, but what was
really interesting about this was that there's this incredible structure to this
and we zoomed in and it looked really cool.
There are period 3 windows, all sorts of complicated behavior in here.
So, there are many values for which the system is chaotic.
The system goes from different period to period in a certain way.
And this has a self-similar structure: it's very complicated but there is some
regularity to this set of behavior for the logistic equation.
So, then we looked at the period doubling route to chaos a little bit more closely.
And in particular I defined this ratio, delta.
It tells us how many times larger branch n is than branch n+1.
So, delta is how much larger or longer this is than that.
That would be delta 1. How much longer, how many times longer is this length
than that? That would be delta 2.
And we looked at the bifurcation diagrams for some different functions,
and I didn't prove it, but we discussed how this quantity, delta, this ratio
of these lengths in the bifurcation diagram is universal.
And that means it has the same value for all functions
provided, a little bit of fine print, they map an interval to itself and have a
single quadratic maximum.
So, this value, which is I believe known to be a rational and I think
transcendental, is known as Feigenbaum's constant, after one of the people who made
this discovery of universality.
This is an amazing mathematical fact and points to some similarities among
a broad class of mathematical systems.
To me, what's even more amazing is that this has physical consequences.
Physical systems show the same universality.
So, the period doubling route to chaos is observed in physical systems.
I talked about a dripping faucet and convection rolls in fluid, and one can
measure delta for these systems. It's not an easy experiment to do,
but it can be done.
And the results are consistent with this universal value, 4.669.
And so, what this tells us is that somehow these simple one-dimensional equations,
we started with a logistic equation, an obviously made-up story about
rabbits on an island, that nevertheless produces a number, a prediction
that you can go out in the real physical world and conduct an experiment
with something much more complicated and get that same number.
So, this is I think one of the most surprising and interesting results
in dynamical systems.
So, then we moved from one-dimensional differential equations to two-dimensional
differential equations.
So now, rather than just keeping track of temperature or population, we're
going to keep track of two populations, say R for rabbits and F for foxes.
And we would have now a system of two coupled differential equations:
the fate of the rabbits depends on rabbits and foxes, and the fate of the
foxes depends on foxes and rabbits.
So, they're coupled, they're linked together.
And one can solve these using Euler's method or things like it,
very, almost identically to how one would for one-dimensional differential equations
And you get two solutions:
you get a rabbit solution and a fox solution.
And in this case, this is the Lotka-Volterra equation,
they both oscillate. We have cycles in both rabbit and foxes.
But then, we could plot R against F.
So, we lose time information, but it will show us how the rabbits and the foxes
are related.
And if we do that, we get a picture that looks like this.
Just a reminder that this curve goes in this direction.
And so, the foxes and rabbits are cycling around.
The rabbit population increases, then the fox population increases.
Rabbits decrease because the foxes are eating them.
Then the foxes decrease because they're sad and hungry because
there aren't rabbits around, and so on.
So, this is is similar to the phase line for one-dimensional equations,
but it's called a phase plane because it lives on a plane.
And this hows how R and F are related.
Phase plane and then phase space is one of the key geometric constructions,
analytical tools used to visualize behavior of dynamical systems.
So, an important result is that there can be no chaos, no aperiodic
solutions in 2D differential equations.
So, curves cannot cross in phase space.
The equations are deterministic, and that means that every point in space, and
remember this is in phase space, so my point in space gives the rabbit and fox
population, there's a unique direction associated with the motion.
DF/DT, DR/DT, that gives a direction. It tells you how the rabbits are
increasing, how the foxes are increasing.
If two phase lines ever cross, like they do where my knuckles are meeting,
then that would be a non-deterministic dynamical system.
There would be two possible trajectories coming from one point.
So, the fact that two curves can't cross in these systems limits the behavior.
They sort of literally paint themselves in as they're tracing something out, tracing
a curve out in phase space.
So there can be stable and unstable fixed points and orbits can tend toward infinity
of course, and there can also be limit cycles attracting cyclic behavior, and we
saw an example of that.
But the main thing is that there can't be aperiodic orbits.
And that result is known as the Poincaré-Bendixson theorem.
It's about a century old.
And it's not immediately obvious; it takes some proof.
Like I said, that's maybe why it's a theorem and not just an obvious statement.
One could imagine, and people in the forums have been trying to imagine
space-filling curves that somehow never repeat but also never leave a bounded area
But the Poincaré-Bendixson theorem says that those solutions somehow
aren't possible.
So, the main result is that two-dimensional differential equations
cannot be chaotic.
That's not the case for three-dimensional differential equations, however.
So, here are the Lorenz equations.
Now, again it's a dynamical system, it's a rule that tells how something changes
in time.
Here that something is x, y, and z, and I forget what parameter values I chose
for sigma, rho, and beta.
And we can get three solutions: x, y, and z.
And these are all curves plotted as a function of time.
But we could plot these in phase space, x, y, and z together.
And for that system, if we do that, we get some complicated structure that
loops around itself and repeats.
It looks like the lines cross, but they don't.
There's actually a space between them.
It looks like they cross because this is a two-dimensional surface trying to
plot something in 3D.
Alright, so just a little bit more about phase space.
Determinism means that curves in phase space cannot intersect.
But because the space is three-dimensional curves can go over or under each other.
And that means that there is a lot more interesting behavior that's possible.
A trajectory can weave around and under and through itself
in some very complicated ways.
What that means, in turn, is that three-dimensional differential equations
can be chaotic.
You can get bounded, aperiodic orbits, and it has sensitive dependence as well.
And then we saw that chaotic trajectories in phase space are particularly
interesting and fun.
They often get pulled to these things called strange attractors.
So here's the Lorenz attractor or the famous values for the Lorenz equation.
Strange attractors. What are strange attractors?
Well, they're attractors, and what that means is that nearby orbits get
pulled into it.
So, if I have a lot of initial conditions, they all are going to get
pulled onto that attractor.
So, in that sense, it's stable. If you're on that attractor and somebody
bumps you off a little bit, you'd get pulled right back towards it.
That's what it means to be stable.
So, it's a stable structure in phase space.
But the motion on the attractor is not periodic the way most attractors that
we've seen are, or even fixed points.
But the motion on the attractor is chaotic.
So, once you're on the attractor, orbits are aperiodic and have
sensitive dependence on initial conditions.
So, it's an attracting chaotic attractor.
Then we looked at this a little bit more geometrically and I argued that the key
ingredients to make a strange attractor or to make chaos of any sort, actually,
is stretching and folding.
So, you need some stretching to pull nearby orbits apart.
The analogy I discussed was kneading dough.
So, when you knead dough, you stretch it.
That pulls things apart, and then you fold it back on itself.
So, the folding keeps orbits bounded.
It takes far apart orbits and moves them closer together.
But stretching pulls nearby orbits apart, and that's what leads to
the butterfly effect, or sensitive dependence.
Now, stretching and folding, it may be relatively easy to picture in
three-dimensional space, either a space of actual dough on a bread board
or a phase space.
But it occurs in one-dimensional maps as well, the logistic equation stretches
and folds.
And this can explain how one-dimensional maps, iterated functions, can capture
some of the features of these higher dimensional systems.
And it begins to explain, also, how these higher dimensional systems,
convection rolls, dripping faucets, can be captured by one-dimensional
functions like the logistic equation and this universal parameter, 4.669.
So, in any event, stretching and folding are the key ingredients for a chaotic
dynamical system.
So, strange attractors once more. They're these complex structures that
arise from simple dynamical systems.
A reminder that we looked at three examples: the Hénon map, the Hénon
attractor, which is a two-dimensional discrete, iterated function.
And then, two different sets of coupled differential equations in three dimensions
the famous Lorenz equations, and also the slightly less famous but equally
beautiful Rössler equations.
Again, the motion on the attractor is chaotic, but all orbits get pulled
to the attractor.
So, strange attractors combine elements of order and disorder.
That's one of the key themes of the course.
The motion on the attractor is locally unstable.
Nearby orbits are getting pulled apart, but globally it's stable.
One has these stable structures, the same Lorenz attractor appears all the time.
If you're on the attractor, you get pushed off it, you get pulled right back in.
Alright, and the last topic we covered in unit 9 was pattern formation.
So, we've seen throughout the course in the first 8 units that dynamical systems
are capable of chaos.
That was one of the main results.
Unpredictable, aperiodic behavior.
But there's a lot more to dynamical systems than chaos.
They can produce patterns, structure, organization, complexity, and so on.
And we looked at just one example of a pattern-forming system.
There are many, many ones to choose from.
But we looked at reaction-diffusion systems.
So there, we have two chemicals that react and diffuse.
And diffusion, that's just the random spreading out of molecules in space,
diffusion tends to smooth out differences, it makes everything
as boring and bland as possible.
But, if we have two different chemicals that react in a certain way,
it's possible to get stable spatial structures even in the presence
of diffusion.
Here are these equations-- I described them in the last unit.
This is deterministic, just like the dynamical systems we've studied before.
And it's spatially extended, because now U and V are functions, not just of T, but
of X and Y.
So, these become partial differential equations.
Crucially, the rule is local.
So, the value of U or the value of V, those are chemical concentrations,
depend on some function of a current value at that location, and on this
Laplacian derivative at that location.
So, we have a local rule in that the chemical concentration here doesn't know
directly what the chemical concentration is here; it's just doing it's own thing
at its own local location.
Nevertheless, it produces these large-scale structures.
So, just one quick example.
We experimented with reaction-diffusion equations at the
Experimentarium Digitale site.
Here's an example that we saw emerging from random initial conditions,
these stable spots appear.
And then we also looked at a video from Stephen Morris at Toronto where two fluids
are poured into this petri dish, and like magic, these patterns start to emerge
out of them.
So, Belousov Zhabotinsky has another example of a reaction-diffusion system.
So, pattern formation is a giant subject.
It could be probably a course in and of itself.
The main point I want to make is that there's more to dynamical systems
than just chaos or unpredictability or irregularity.
Simple, spatially-extended dynamical systems with local rules are capable
of producing stable, global patterns and structures.
So, there's a lot more to the study of chaos than chaos.
Simple dynamical systems can produce complexity and all sorts of interesting
emergent structures and phenomena.