0:00:02.720,0:00:04.372
Continuing with our review,
0:00:04.372,0:00:09.852
the next topics that we covered concern[br]bifurcation diagrams.
0:00:09.872,0:00:14.190
They're a way to see how the behavior of[br]a dynamical system changes as a parameter
0:00:14.190,0:00:15.270
is changed.
0:00:15.270,0:00:20.220
I think it's best to think of them as[br]being built up one parameter value
0:00:20.220,0:00:21.510
at a time.
0:00:21.510,0:00:25.510
So, for each parameter value, make a phase[br]line if it's a differential equation,
0:00:25.510,0:00:28.120
or a final-state diagram for an[br]iterated function.
0:00:28.120,0:00:32.120
And you get a collection of these, and[br]then you glue these together to make a
0:00:32.120,0:00:34.550
bifurcation diagram.
0:00:34.550,0:00:37.090
So, here's one of the first bifurcation[br]diagrams we looked at.
0:00:37.090,0:00:41.012
This is the logistic equation with[br]harvest. The equation is down here.
0:00:41.012,0:00:45.012
And so, h is the parameter that [br]I'm changing.
0:00:45.012,0:00:56.532
H is here, it goes from 0 to 100 to 200[br]and so on.
0:00:56.532,0:01:01.449
And so, a way to interpret this is [br]suppose you want to know what's going on
0:01:01.449,0:01:09.239
at h is 100. Well, I would try to focus[br]right on that value, and I can see "aha,"
0:01:09.239,0:01:13.379
it looks to me like there is an attracting[br]fixed point here, and a repelling
0:01:13.379,0:01:18.649
fixed point there. So there are two fixed[br]points: one of them attracting and one of
0:01:18.649,0:01:21.789
them repelling, or repulsive.
0:01:21.789,0:01:25.519
And, what's interesting about this is that[br]so this is is the stable fixed point,
0:01:25.519,0:01:30.609
this would be the stable population of [br]the story I told involved fish in a lake
0:01:30.609,0:01:35.139
or an ocean, and h is the fishing rate,[br]how many fish you catch every year.
0:01:35.139,0:01:41.619
And that increases, and as you increase h,[br]the sort of steady state population of the
0:01:41.619,0:01:43.879
fish decreases, that makes sense.
0:01:43.879,0:01:48.539
But what's surprising is that when [br]you're here, and you make a tiny increase
0:01:48.539,0:01:56.039
in the fishing rate, the steady-state[br]population crashes and in fact disappears.
0:01:56.039,0:01:57.627
The population crashes.
0:01:57.627,0:02:03.437
So, you have a small change in h leading[br]to a very large qualitative change in the
0:02:03.437,0:02:04.657
fish behavior.
0:02:04.657,0:02:08.007
So, this is an example of a bifurcation [br]that occurs right here.
0:02:08.007,0:02:12.007
It's a sudden qualitative change in the [br]system's behavior as a parameter
0:02:12.007,0:02:15.647
is varied slowly and continuously.
0:02:15.647,0:02:19.647
So, we looked at bifurcation diagrams for[br]differential equations and we saw the
0:02:19.647,0:02:25.187
surprising discontinuous behavior, then we[br]looked at bifurcation diagrams for the
0:02:25.187,0:02:30.837
logistic equation, and we saw bifurcations[br]here is period 2 to period 4, but what was
0:02:30.837,0:02:34.837
really interesting about this was that[br]there's this incredible structure to this
0:02:34.837,0:02:37.077
and we zoomed in and it looked really[br]cool.
0:02:37.077,0:02:41.077
There are period 3 windows, all sorts of[br]complicated behavior in here.
0:02:41.077,0:02:45.077
So, there are many values for which the[br]system is chaotic.
0:02:45.077,0:02:49.077
The system goes from different period [br]to period in a certain way.
0:02:49.077,0:02:54.817
And this has a self-similar structure:[br]it's very complicated but there is some
0:02:54.817,0:03:01.367
regularity to this set of behavior for the[br]logistic equation.
0:03:01.367,0:03:06.117
So, then we looked at the period doubling[br]route to chaos a little bit more closely.
0:03:06.117,0:03:09.127
And in particular I defined this ratio,[br]delta.
0:03:09.127,0:03:13.127
It tells us how many times larger branch n[br]is than branch n+1.
0:03:13.127,0:03:18.927
So, delta is how much larger or longer[br]this is than that.
0:03:18.927,0:03:22.317
That would be delta 1. How much longer,[br]how many times longer is this length
0:03:22.317,0:03:26.617
than that? [br]That would be delta 2.
0:03:26.617,0:03:31.687
And we looked at the bifurcation diagrams[br]for some different functions,
0:03:31.687,0:03:39.397
and I didn't prove it, but we discussed [br]how this quantity, delta, this ratio
0:03:39.397,0:03:43.237
of these lengths in the bifurcation[br]diagram is universal.
0:03:43.237,0:03:46.237
And that means it has the same value for[br]all functions
0:03:46.237,0:03:49.707
provided, a little bit of fine print,[br]they map an interval to itself and have a
0:03:49.707,0:03:51.627
single quadratic maximum.
0:03:51.627,0:03:56.347
So, this value, which is I believe [br]known to be a rational and I think
0:03:56.347,0:04:02.837
transcendental, is known as Feigenbaum's [br]constant, after one of the people who made
0:04:02.837,0:04:05.187
this discovery of universality.
0:04:05.187,0:04:10.257
This is an amazing mathematical fact[br]and points to some similarities among
0:04:10.257,0:04:13.157
a broad class of mathematical systems.
0:04:13.157,0:04:17.157
To me, what's even more amazing is that[br]this has physical consequences.
0:04:17.157,0:04:21.017
Physical systems show the same[br]universality.
0:04:21.017,0:04:24.577
So, the period doubling route to chaos[br]is observed in physical systems.
0:04:24.577,0:04:28.577
I talked about a dripping faucet and [br]convection rolls in fluid, and one can
0:04:28.577,0:04:31.287
measure delta for these systems.[br]It's not an easy experiment to do,
0:04:31.287,0:04:32.767
but it can be done.
0:04:32.767,0:04:37.777
And the results are consistent with this[br]universal value, 4.669.
0:04:37.777,0:04:43.407
And so, what this tells us is that somehow[br]these simple one-dimensional equations,
0:04:43.407,0:04:46.067
we started with a logistic equation,[br]an obviously made-up story about
0:04:46.067,0:04:51.357
rabbits on an island, that nevertheless[br]produces a number, a prediction
0:04:51.357,0:04:55.357
that you can go out in the real physical[br]world and conduct an experiment
0:04:55.357,0:04:59.357
with something much more complicated[br]and get that same number.
0:04:59.357,0:05:03.357
So, this is I think one of the most [br]surprising and interesting results
0:05:03.357,0:05:08.297
in dynamical systems.
0:05:08.297,0:05:12.087
So, then we moved from one-dimensional[br]differential equations to two-dimensional
0:05:12.087,0:05:13.767
differential equations.
0:05:13.767,0:05:17.337
So now, rather than just keeping track[br]of temperature or population, we're
0:05:17.337,0:05:22.607
going to keep track of two populations,[br]say R for rabbits and F for foxes.
0:05:22.607,0:05:26.607
And we would have now a system of two[br]coupled differential equations:
0:05:26.607,0:05:30.357
the fate of the rabbits depends on [br]rabbits and foxes, and the fate of the
0:05:30.357,0:05:33.477
foxes depends on foxes and rabbits.[br]
0:05:33.477,0:05:35.827
So, they're coupled, they're [br]linked together.
0:05:35.827,0:05:39.827
And one can solve these using Euler's [br]method or things like it,
0:05:39.827,0:05:44.457
very, almost identically to how one would[br]for one-dimensional differential equations
0:05:44.457,0:05:45.837
And you get two solutions:
0:05:45.837,0:05:47.657
you get a rabbit solution and a fox[br]solution.
0:05:47.657,0:05:51.657
And in this case, this is the [br]Lotka-Volterra equation,
0:05:51.657,0:05:55.657
they both oscillate.[br]We have cycles in both rabbit and foxes.
0:05:55.657,0:05:59.657
But then, we could plot R against F.
0:05:59.657,0:06:03.657
So, we lose time information, but it will[br]show us how the rabbits and the foxes
0:06:03.657,0:06:05.837
are related.
0:06:05.837,0:06:09.837
And if we do that, we get a picture that [br]looks like this.
0:06:09.837,0:06:13.837
Just a reminder that this curve goes in [br]this direction.
0:06:13.837,0:06:19.747
And so, the foxes and rabbits are [br]cycling around.
0:06:19.747,0:06:23.747
The rabbit population increases,[br]then the fox population increases.
0:06:23.747,0:06:27.747
Rabbits decrease because the foxes[br]are eating them.
0:06:27.747,0:06:31.107
Then the foxes decrease because [br]they're sad and hungry because
0:06:31.107,0:06:33.527
there aren't rabbits around, and so on.
0:06:33.527,0:06:37.077
So, this is is similar to the phase line [br]for one-dimensional equations,
0:06:37.077,0:06:40.307
but it's called a phase plane because it [br]lives on a plane.
0:06:40.307,0:06:43.337
And this hows how R and F are related.
0:06:43.337,0:06:47.337
Phase plane and then phase space is [br]one of the key geometric constructions,
0:06:47.337,0:06:53.867
analytical tools used to visualize [br]behavior of dynamical systems.
0:06:53.867,0:06:59.397
So, an important result is that there[br]can be no chaos, no aperiodic
0:06:59.397,0:07:04.037
solutions in 2D differential equations.
0:07:04.037,0:07:08.317
So, curves cannot cross in phase space.
0:07:08.317,0:07:12.317
The equations are deterministic, and that[br]means that every point in space, and
0:07:12.317,0:07:16.317
remember this is in phase space, so my[br]point in space gives the rabbit and fox
0:07:16.317,0:07:24.197
population, there's a unique direction[br]associated with the motion.
0:07:24.197,0:07:28.267
DF/DT, DR/DT, that gives a direction.[br]It tells you how the rabbits are
0:07:28.267,0:07:30.527
increasing, how the foxes are increasing.
0:07:30.527,0:07:36.187
If two phase lines ever cross, like they[br]do where my knuckles are meeting,
0:07:36.187,0:07:39.707
then that would be a non-deterministic[br]dynamical system.
0:07:39.707,0:07:43.707
There would be two possible trajectories[br]coming from one point.
0:07:43.707,0:07:49.147
So, the fact that two curves can't cross[br]in these systems limits the behavior.
0:07:49.147,0:07:53.147
They sort of literally paint themselves in[br]as they're tracing something out, tracing
0:07:53.147,0:07:57.147
a curve out in phase space.
0:07:57.147,0:08:01.147
So there can be stable and unstable fixed[br]points and orbits can tend toward infinity
0:08:01.147,0:08:04.927
of course, and there can also be limit[br]cycles attracting cyclic behavior, and we
0:08:04.927,0:08:06.537
saw an example of that.
0:08:06.537,0:08:09.447
But the main thing is that there can't be [br]aperiodic orbits.
0:08:09.447,0:08:12.967
And that result is known as the [br]Poincaré-Bendixson theorem.
0:08:12.967,0:08:15.807
It's about a century old.
0:08:15.807,0:08:19.807
And it's not immediately obvious;[br]it takes some proof.
0:08:19.807,0:08:24.907
Like I said, that's maybe why it's a [br]theorem and not just an obvious statement.
0:08:24.907,0:08:29.517
One could imagine, and people in the[br]forums have been trying to imagine
0:08:29.517,0:08:39.337
space-filling curves that somehow never[br]repeat but also never leave a bounded area
0:08:39.337,0:08:44.697
But the Poincaré-Bendixson theorem says [br]that those solutions somehow
0:08:44.697,0:08:46.167
aren't possible.
0:08:46.167,0:08:48.887
So, the main result is that[br]two-dimensional differential equations
0:08:48.887,0:08:51.907
cannot be chaotic.
0:08:51.907,0:08:54.577
That's not the case for three-dimensional[br]differential equations, however.
0:08:54.577,0:08:56.917
So, here are the Lorenz equations.
0:08:56.917,0:09:00.147
Now, again it's a dynamical system, it's[br]a rule that tells how something changes
0:09:00.147,0:09:01.247
in time.
0:09:01.247,0:09:08.167
Here that something is x, y, and z, and[br]I forget what parameter values I chose
0:09:08.167,0:09:09.597
for sigma, rho, and beta.
0:09:09.597,0:09:12.687
And we can get three solutions:[br]x, y, and z.
0:09:12.687,0:09:15.297
And these are all curves plotted as a [br]function of time.
0:09:15.297,0:09:19.887
But we could plot these in phase space,[br]x, y, and z together.
0:09:19.887,0:09:24.117
And for that system, if we do that, we get[br]some complicated structure that
0:09:24.117,0:09:27.427
loops around itself and repeats.
0:09:27.427,0:09:30.517
It looks like the lines cross, but [br]they don't.
0:09:30.517,0:09:33.037
There's actually a space between them.
0:09:33.037,0:09:36.247
It looks like they cross because this is[br]a two-dimensional surface trying to
0:09:36.247,0:09:39.697
plot something in 3D.
0:09:39.697,0:09:46.587
Alright, so just a little bit more about[br]phase space.
0:09:46.587,0:09:49.817
Determinism means that curves in phase[br]space cannot intersect.
0:09:49.817,0:09:53.817
But because the space is three-dimensional[br]curves can go over or under each other.
0:09:53.817,0:09:58.577
And that means that there is a lot more [br]interesting behavior that's possible.
0:09:58.577,0:10:02.577
A trajectory can weave around and under[br]and through itself
0:10:02.577,0:10:04.557
in some very complicated ways.
0:10:04.557,0:10:08.057
What that means, in turn, is that [br]three-dimensional differential equations
0:10:08.057,0:10:09.367
can be chaotic.
0:10:09.367,0:10:16.047
You can get bounded, aperiodic orbits,[br]and it has sensitive dependence as well.
0:10:16.047,0:10:19.367
And then we saw that chaotic trajectories[br]in phase space are particularly
0:10:19.367,0:10:20.847
interesting and fun.
0:10:20.847,0:10:24.247
They often get pulled to these things[br]called strange attractors.
0:10:24.247,0:10:30.457
So here's the Lorenz attractor or the[br]famous values for the Lorenz equation.
0:10:30.457,0:10:32.737
Strange attractors.[br]What are strange attractors?
0:10:32.737,0:10:36.737
Well, they're attractors, and what that [br]means is that nearby orbits get
0:10:36.737,0:10:38.017
pulled into it.
0:10:38.017,0:10:41.537
So, if I have a lot of initial conditions,[br]they all are going to get
0:10:41.537,0:10:43.177
pulled onto that attractor.
0:10:43.177,0:10:47.177
So, in that sense, it's stable.[br]If you're on that attractor and somebody
0:10:47.177,0:10:50.337
bumps you off a little bit, you'd get[br]pulled right back towards it.
0:10:50.337,0:10:52.057
That's what it means to be stable.
0:10:52.057,0:10:54.797
So, it's a stable structure [br]in phase space.
0:10:54.797,0:10:58.797
But the motion on the attractor is not[br]periodic the way most attractors that
0:10:58.797,0:11:01.027
we've seen are, or even fixed points.[br]
0:11:01.027,0:11:03.697
But the motion on the attractor [br]is chaotic.
0:11:03.697,0:11:07.047
So, once you're on the attractor, orbits[br]are aperiodic and have
0:11:07.047,0:11:09.387
sensitive dependence on[br]initial conditions.
0:11:09.387,0:11:12.227
So, it's an attracting chaotic attractor.
0:11:16.043,0:11:20.043
Then we looked at this a little bit more[br]geometrically and I argued that the key
0:11:20.043,0:11:23.603
ingredients to make a strange attractor[br]or to make chaos of any sort, actually,
0:11:23.603,0:11:25.243
is stretching and folding.
0:11:25.243,0:11:28.833
So, you need some stretching to pull[br]nearby orbits apart.
0:11:28.833,0:11:32.833
The analogy I discussed was[br]kneading dough.
0:11:32.833,0:11:35.613
So, when you knead dough, you stretch it.
0:11:35.613,0:11:39.613
That pulls things apart, and then you fold[br]it back on itself.
0:11:39.613,0:11:41.473
So, the folding keeps orbits bounded.
0:11:41.473,0:11:44.903
It takes far apart orbits and moves them[br]closer together.
0:11:44.903,0:11:47.473
But stretching pulls nearby orbits apart,[br]and that's what leads to
0:11:47.473,0:11:50.683
the butterfly effect,[br]or sensitive dependence.
0:11:50.683,0:11:55.433
Now, stretching and folding, it may be [br]relatively easy to picture in
0:11:55.433,0:11:59.433
three-dimensional space, either a space[br]of actual dough on a bread board
0:11:59.433,0:12:01.143
or a phase space.
0:12:01.143,0:12:04.653
But it occurs in one-dimensional maps[br]as well, the logistic equation stretches
0:12:04.653,0:12:06.103
and folds.
0:12:06.103,0:12:10.103
And this can explain how one-dimensional[br]maps, iterated functions, can capture
0:12:10.103,0:12:14.103
some of the features of these[br]higher dimensional systems.
0:12:14.103,0:12:21.723
And it begins to explain, also, how these[br]higher dimensional systems,
0:12:21.723,0:12:26.733
convection rolls, dripping faucets,[br]can be captured by one-dimensional
0:12:26.733,0:12:31.563
functions like the logistic equation and[br]this universal parameter, 4.669.
0:12:31.563,0:12:36.263
So, in any event, stretching and folding [br]are the key ingredients for a chaotic
0:12:36.263,0:12:38.473
dynamical system.
0:12:38.473,0:12:43.533
So, strange attractors once more. [br]They're these complex structures that
0:12:43.533,0:12:45.783
arise from simple dynamical systems.
0:12:45.783,0:12:48.583
A reminder that we looked at three[br]examples: the Hénon map, the Hénon
0:12:48.583,0:12:54.473
attractor, which is a two-dimensional[br]discrete, iterated function.
0:12:54.473,0:13:00.093
And then, two different sets of coupled [br]differential equations in three dimensions
0:13:00.093,0:13:04.483
the famous Lorenz equations, and also[br]the slightly less famous but equally
0:13:04.483,0:13:06.883
beautiful Rössler equations.
0:13:06.883,0:13:10.643
Again, the motion on the attractor is[br]chaotic, but all orbits get pulled
0:13:10.643,0:13:11.623
to the attractor.
0:13:11.623,0:13:15.623
So, strange attractors combine elements[br]of order and disorder.
0:13:15.623,0:13:18.163
That's one of the key themes [br]of the course.
0:13:18.163,0:13:20.883
The motion on the attractor is locally[br]unstable.
0:13:20.883,0:13:24.883
Nearby orbits are getting pulled apart,[br]but globally it's stable.
0:13:24.883,0:13:31.883
One has these stable structures, the same[br]Lorenz attractor appears all the time.
0:13:31.883,0:13:35.383
If you're on the attractor, you get pushed[br]off it, you get pulled right back in.
0:13:37.816,0:13:41.816
Alright, and the last topic we covered in [br]unit 9 was pattern formation.
0:13:41.816,0:13:45.056
So, we've seen throughout the course in[br]the first 8 units that dynamical systems
0:13:45.056,0:13:46.886
are capable of chaos.
0:13:46.886,0:13:49.106
That was one of the main results.
0:13:49.106,0:13:51.296
Unpredictable, aperiodic behavior.
0:13:51.296,0:13:54.246
But there's a lot more to dynamical[br]systems than chaos.
0:13:54.246,0:13:58.246
They can produce patterns, structure,[br]organization, complexity, and so on.
0:13:58.246,0:14:01.096
And we looked at just one example[br]of a pattern-forming system.
0:14:01.096,0:14:03.276
There are many, many ones to choose from.
0:14:03.276,0:14:05.686
But we looked at reaction-diffusion[br]systems.
0:14:07.491,0:14:10.301
So there, we have two chemicals that [br]react and diffuse.
0:14:10.687,0:14:15.737
And diffusion, that's just the random[br]spreading out of molecules in space,
0:14:15.737,0:14:19.617
diffusion tends to smooth out [br]differences, it makes everything
0:14:19.617,0:14:22.317
as boring and bland as possible.
0:14:22.317,0:14:26.317
But, if we have two different chemicals[br]that react in a certain way,
0:14:26.317,0:14:29.117
it's possible to get stable spatial [br]structures even in the presence
0:14:29.117,0:14:30.467
of diffusion.
0:14:30.467,0:14:33.617
Here are these equations-- I described[br]them in the last unit.
0:14:33.617,0:14:38.047
This is deterministic, just like the [br]dynamical systems we've studied before.
0:14:38.047,0:14:42.617
And it's spatially extended, because now[br]U and V are functions, not just of T, but
0:14:42.617,0:14:44.157
of X and Y.
0:14:44.157,0:14:47.387
So, these become partial differential[br]equations.
0:14:47.387,0:14:49.867
Crucially, the rule is local.
0:14:49.867,0:14:55.067
So, the value of U or the value of V, [br]those are chemical concentrations,
0:14:55.067,0:14:59.817
depend on some function of a current[br]value at that location, and on this
0:14:59.817,0:15:03.527
Laplacian derivative at that location.
0:15:03.527,0:15:10.047
So, we have a local rule in that the [br]chemical concentration here doesn't know
0:15:10.047,0:15:13.957
directly what the chemical concentration[br]is here; it's just doing it's own thing
0:15:13.957,0:15:15.877
at its own local location.
0:15:15.877,0:15:19.897
Nevertheless, it produces these [br]large-scale structures.
0:15:20.958,0:15:23.678
So, just one quick example.
0:15:23.678,0:15:26.468
We experimented with reaction-diffusion[br]equations at the
0:15:26.468,0:15:28.828
Experimentarium Digitale site.
0:15:28.828,0:15:31.948
Here's an example that we saw emerging[br]from random initial conditions,
0:15:31.948,0:15:35.168
these stable spots appear.
0:15:35.168,0:15:41.828
And then we also looked at a video from[br]Stephen Morris at Toronto where two fluids
0:15:41.828,0:15:46.661
are poured into this petri dish, and like[br]magic, these patterns start to emerge
0:15:46.661,0:15:47.761
out of them.
0:15:47.761,0:15:51.441
So, Belousov Zhabotinsky has another[br]example of a reaction-diffusion system.
0:15:53.243,0:15:56.153
So, pattern formation is a giant subject.
0:15:56.153,0:15:58.833
It could be probably a course in and of[br]itself.
0:15:58.833,0:16:01.703
The main point I want to make is that[br]there's more to dynamical systems
0:16:01.703,0:16:04.963
than just chaos or unpredictability[br]or irregularity.
0:16:04.963,0:16:09.213
Simple, spatially-extended dynamical[br]systems with local rules are capable
0:16:09.213,0:16:13.213
of producing stable, global patterns and[br]structures.
0:16:13.213,0:16:16.673
So, there's a lot more to the study of[br]chaos than chaos.
0:16:16.673,0:16:20.673
Simple dynamical systems can produce[br]complexity and all sorts of interesting
0:16:20.673,0:16:22.983
emergent structures and phenomena.