[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:02.72,0:00:04.37,Default,,0000,0000,0000,,Continuing with our review,
Dialogue: 0,0:00:04.37,0:00:09.85,Default,,0000,0000,0000,,the next topics that we covered concern\Nbifurcation diagrams.
Dialogue: 0,0:00:09.87,0:00:14.19,Default,,0000,0000,0000,,They're a way to see how the behavior of\Na dynamical system changes as a parameter
Dialogue: 0,0:00:14.19,0:00:15.27,Default,,0000,0000,0000,,is changed.
Dialogue: 0,0:00:15.27,0:00:20.22,Default,,0000,0000,0000,,I think it's best to think of them as\Nbeing built up one parameter value
Dialogue: 0,0:00:20.22,0:00:21.51,Default,,0000,0000,0000,,at a time.
Dialogue: 0,0:00:21.51,0:00:25.51,Default,,0000,0000,0000,,So, for each parameter value, make a phase\Nline if it's a differential equation,
Dialogue: 0,0:00:25.51,0:00:28.12,Default,,0000,0000,0000,,or a final-state diagram for an\Niterated function.
Dialogue: 0,0:00:28.12,0:00:32.12,Default,,0000,0000,0000,,And you get a collection of these, and\Nthen you glue these together to make a
Dialogue: 0,0:00:32.12,0:00:34.55,Default,,0000,0000,0000,,bifurcation diagram.
Dialogue: 0,0:00:34.55,0:00:37.09,Default,,0000,0000,0000,,So, here's one of the first bifurcation\Ndiagrams we looked at.
Dialogue: 0,0:00:37.09,0:00:41.01,Default,,0000,0000,0000,,This is the logistic equation with\Nharvest. The equation is down here.
Dialogue: 0,0:00:41.01,0:00:45.01,Default,,0000,0000,0000,,And so, h is the parameter that \NI'm changing.
Dialogue: 0,0:00:45.01,0:00:56.53,Default,,0000,0000,0000,,H is here, it goes from 0 to 100 to 200\Nand so on.
Dialogue: 0,0:00:56.53,0:01:01.45,Default,,0000,0000,0000,,And so, a way to interpret this is \Nsuppose you want to know what's going on
Dialogue: 0,0:01:01.45,0:01:09.24,Default,,0000,0000,0000,,at h is 100. Well, I would try to focus\Nright on that value, and I can see "aha,"
Dialogue: 0,0:01:09.24,0:01:13.38,Default,,0000,0000,0000,,it looks to me like there is an attracting\Nfixed point here, and a repelling
Dialogue: 0,0:01:13.38,0:01:18.65,Default,,0000,0000,0000,,fixed point there. So there are two fixed\Npoints: one of them attracting and one of
Dialogue: 0,0:01:18.65,0:01:21.79,Default,,0000,0000,0000,,them repelling, or repulsive.
Dialogue: 0,0:01:21.79,0:01:25.52,Default,,0000,0000,0000,,And, what's interesting about this is that\Nso this is is the stable fixed point,
Dialogue: 0,0:01:25.52,0:01:30.61,Default,,0000,0000,0000,,this would be the stable population of \Nthe story I told involved fish in a lake
Dialogue: 0,0:01:30.61,0:01:35.14,Default,,0000,0000,0000,,or an ocean, and h is the fishing rate,\Nhow many fish you catch every year.
Dialogue: 0,0:01:35.14,0:01:41.62,Default,,0000,0000,0000,,And that increases, and as you increase h,\Nthe sort of steady state population of the
Dialogue: 0,0:01:41.62,0:01:43.88,Default,,0000,0000,0000,,fish decreases, that makes sense.
Dialogue: 0,0:01:43.88,0:01:48.54,Default,,0000,0000,0000,,But what's surprising is that when \Nyou're here, and you make a tiny increase
Dialogue: 0,0:01:48.54,0:01:56.04,Default,,0000,0000,0000,,in the fishing rate, the steady-state\Npopulation crashes and in fact disappears.
Dialogue: 0,0:01:56.04,0:01:57.63,Default,,0000,0000,0000,,The population crashes.
Dialogue: 0,0:01:57.63,0:02:03.44,Default,,0000,0000,0000,,So, you have a small change in h leading\Nto a very large qualitative change in the
Dialogue: 0,0:02:03.44,0:02:04.66,Default,,0000,0000,0000,,fish behavior.
Dialogue: 0,0:02:04.66,0:02:08.01,Default,,0000,0000,0000,,So, this is an example of a bifurcation \Nthat occurs right here.
Dialogue: 0,0:02:08.01,0:02:12.01,Default,,0000,0000,0000,,It's a sudden qualitative change in the \Nsystem's behavior as a parameter
Dialogue: 0,0:02:12.01,0:02:15.65,Default,,0000,0000,0000,,is varied slowly and continuously.
Dialogue: 0,0:02:15.65,0:02:19.65,Default,,0000,0000,0000,,So, we looked at bifurcation diagrams for\Ndifferential equations and we saw the
Dialogue: 0,0:02:19.65,0:02:25.19,Default,,0000,0000,0000,,surprising discontinuous behavior, then we\Nlooked at bifurcation diagrams for the
Dialogue: 0,0:02:25.19,0:02:30.84,Default,,0000,0000,0000,,logistic equation, and we saw bifurcations\Nhere is period 2 to period 4, but what was
Dialogue: 0,0:02:30.84,0:02:34.84,Default,,0000,0000,0000,,really interesting about this was that\Nthere's this incredible structure to this
Dialogue: 0,0:02:34.84,0:02:37.08,Default,,0000,0000,0000,,and we zoomed in and it looked really\Ncool.
Dialogue: 0,0:02:37.08,0:02:41.08,Default,,0000,0000,0000,,There are period 3 windows, all sorts of\Ncomplicated behavior in here.
Dialogue: 0,0:02:41.08,0:02:45.08,Default,,0000,0000,0000,,So, there are many values for which the\Nsystem is chaotic.
Dialogue: 0,0:02:45.08,0:02:49.08,Default,,0000,0000,0000,,The system goes from different period \Nto period in a certain way.
Dialogue: 0,0:02:49.08,0:02:54.82,Default,,0000,0000,0000,,And this has a self-similar structure:\Nit's very complicated but there is some
Dialogue: 0,0:02:54.82,0:03:01.37,Default,,0000,0000,0000,,regularity to this set of behavior for the\Nlogistic equation.
Dialogue: 0,0:03:01.37,0:03:06.12,Default,,0000,0000,0000,,So, then we looked at the period doubling\Nroute to chaos a little bit more closely.
Dialogue: 0,0:03:06.12,0:03:09.13,Default,,0000,0000,0000,,And in particular I defined this ratio,\Ndelta.
Dialogue: 0,0:03:09.13,0:03:13.13,Default,,0000,0000,0000,,It tells us how many times larger branch n\Nis than branch n+1.
Dialogue: 0,0:03:13.13,0:03:18.93,Default,,0000,0000,0000,,So, delta is how much larger or longer\Nthis is than that.
Dialogue: 0,0:03:18.93,0:03:22.32,Default,,0000,0000,0000,,That would be delta 1. How much longer,\Nhow many times longer is this length
Dialogue: 0,0:03:22.32,0:03:26.62,Default,,0000,0000,0000,,than that? \NThat would be delta 2.
Dialogue: 0,0:03:26.62,0:03:31.69,Default,,0000,0000,0000,,And we looked at the bifurcation diagrams\Nfor some different functions,
Dialogue: 0,0:03:31.69,0:03:39.40,Default,,0000,0000,0000,,and I didn't prove it, but we discussed \Nhow this quantity, delta, this ratio
Dialogue: 0,0:03:39.40,0:03:43.24,Default,,0000,0000,0000,,of these lengths in the bifurcation\Ndiagram is universal.
Dialogue: 0,0:03:43.24,0:03:46.24,Default,,0000,0000,0000,,And that means it has the same value for\Nall functions
Dialogue: 0,0:03:46.24,0:03:49.71,Default,,0000,0000,0000,,provided, a little bit of fine print,\Nthey map an interval to itself and have a
Dialogue: 0,0:03:49.71,0:03:51.63,Default,,0000,0000,0000,,single quadratic maximum.
Dialogue: 0,0:03:51.63,0:03:56.35,Default,,0000,0000,0000,,So, this value, which is I believe \Nknown to be a rational and I think
Dialogue: 0,0:03:56.35,0:04:02.84,Default,,0000,0000,0000,,transcendental, is known as Feigenbaum's \Nconstant, after one of the people who made
Dialogue: 0,0:04:02.84,0:04:05.19,Default,,0000,0000,0000,,this discovery of universality.
Dialogue: 0,0:04:05.19,0:04:10.26,Default,,0000,0000,0000,,This is an amazing mathematical fact\Nand points to some similarities among
Dialogue: 0,0:04:10.26,0:04:13.16,Default,,0000,0000,0000,,a broad class of mathematical systems.
Dialogue: 0,0:04:13.16,0:04:17.16,Default,,0000,0000,0000,,To me, what's even more amazing is that\Nthis has physical consequences.
Dialogue: 0,0:04:17.16,0:04:21.02,Default,,0000,0000,0000,,Physical systems show the same\Nuniversality.
Dialogue: 0,0:04:21.02,0:04:24.58,Default,,0000,0000,0000,,So, the period doubling route to chaos\Nis observed in physical systems.
Dialogue: 0,0:04:24.58,0:04:28.58,Default,,0000,0000,0000,,I talked about a dripping faucet and \Nconvection rolls in fluid, and one can
Dialogue: 0,0:04:28.58,0:04:31.29,Default,,0000,0000,0000,,measure delta for these systems.\NIt's not an easy experiment to do,
Dialogue: 0,0:04:31.29,0:04:32.77,Default,,0000,0000,0000,,but it can be done.
Dialogue: 0,0:04:32.77,0:04:37.78,Default,,0000,0000,0000,,And the results are consistent with this\Nuniversal value, 4.669.
Dialogue: 0,0:04:37.78,0:04:43.41,Default,,0000,0000,0000,,And so, what this tells us is that somehow\Nthese simple one-dimensional equations,
Dialogue: 0,0:04:43.41,0:04:46.07,Default,,0000,0000,0000,,we started with a logistic equation,\Nan obviously made-up story about
Dialogue: 0,0:04:46.07,0:04:51.36,Default,,0000,0000,0000,,rabbits on an island, that nevertheless\Nproduces a number, a prediction
Dialogue: 0,0:04:51.36,0:04:55.36,Default,,0000,0000,0000,,that you can go out in the real physical\Nworld and conduct an experiment
Dialogue: 0,0:04:55.36,0:04:59.36,Default,,0000,0000,0000,,with something much more complicated\Nand get that same number.
Dialogue: 0,0:04:59.36,0:05:03.36,Default,,0000,0000,0000,,So, this is I think one of the most \Nsurprising and interesting results
Dialogue: 0,0:05:03.36,0:05:08.30,Default,,0000,0000,0000,,in dynamical systems.
Dialogue: 0,0:05:08.30,0:05:12.09,Default,,0000,0000,0000,,So, then we moved from one-dimensional\Ndifferential equations to two-dimensional
Dialogue: 0,0:05:12.09,0:05:13.77,Default,,0000,0000,0000,,differential equations.
Dialogue: 0,0:05:13.77,0:05:17.34,Default,,0000,0000,0000,,So now, rather than just keeping track\Nof temperature or population, we're
Dialogue: 0,0:05:17.34,0:05:22.61,Default,,0000,0000,0000,,going to keep track of two populations,\Nsay R for rabbits and F for foxes.
Dialogue: 0,0:05:22.61,0:05:26.61,Default,,0000,0000,0000,,And we would have now a system of two\Ncoupled differential equations:
Dialogue: 0,0:05:26.61,0:05:30.36,Default,,0000,0000,0000,,the fate of the rabbits depends on \Nrabbits and foxes, and the fate of the
Dialogue: 0,0:05:30.36,0:05:33.48,Default,,0000,0000,0000,,foxes depends on foxes and rabbits.\N
Dialogue: 0,0:05:33.48,0:05:35.83,Default,,0000,0000,0000,,So, they're coupled, they're \Nlinked together.
Dialogue: 0,0:05:35.83,0:05:39.83,Default,,0000,0000,0000,,And one can solve these using Euler's \Nmethod or things like it,
Dialogue: 0,0:05:39.83,0:05:44.46,Default,,0000,0000,0000,,very, almost identically to how one would\Nfor one-dimensional differential equations
Dialogue: 0,0:05:44.46,0:05:45.84,Default,,0000,0000,0000,,And you get two solutions:
Dialogue: 0,0:05:45.84,0:05:47.66,Default,,0000,0000,0000,,you get a rabbit solution and a fox\Nsolution.
Dialogue: 0,0:05:47.66,0:05:51.66,Default,,0000,0000,0000,,And in this case, this is the \NLotka-Volterra equation,
Dialogue: 0,0:05:51.66,0:05:55.66,Default,,0000,0000,0000,,they both oscillate.\NWe have cycles in both rabbit and foxes.
Dialogue: 0,0:05:55.66,0:05:59.66,Default,,0000,0000,0000,,But then, we could plot R against F.
Dialogue: 0,0:05:59.66,0:06:03.66,Default,,0000,0000,0000,,So, we lose time information, but it will\Nshow us how the rabbits and the foxes
Dialogue: 0,0:06:03.66,0:06:05.84,Default,,0000,0000,0000,,are related.
Dialogue: 0,0:06:05.84,0:06:09.84,Default,,0000,0000,0000,,And if we do that, we get a picture that \Nlooks like this.
Dialogue: 0,0:06:09.84,0:06:13.84,Default,,0000,0000,0000,,Just a reminder that this curve goes in \Nthis direction.
Dialogue: 0,0:06:13.84,0:06:19.75,Default,,0000,0000,0000,,And so, the foxes and rabbits are \Ncycling around.
Dialogue: 0,0:06:19.75,0:06:23.75,Default,,0000,0000,0000,,The rabbit population increases,\Nthen the fox population increases.
Dialogue: 0,0:06:23.75,0:06:27.75,Default,,0000,0000,0000,,Rabbits decrease because the foxes\Nare eating them.
Dialogue: 0,0:06:27.75,0:06:31.11,Default,,0000,0000,0000,,Then the foxes decrease because \Nthey're sad and hungry because
Dialogue: 0,0:06:31.11,0:06:33.53,Default,,0000,0000,0000,,there aren't rabbits around, and so on.
Dialogue: 0,0:06:33.53,0:06:37.08,Default,,0000,0000,0000,,So, this is is similar to the phase line \Nfor one-dimensional equations,
Dialogue: 0,0:06:37.08,0:06:40.31,Default,,0000,0000,0000,,but it's called a phase plane because it \Nlives on a plane.
Dialogue: 0,0:06:40.31,0:06:43.34,Default,,0000,0000,0000,,And this hows how R and F are related.
Dialogue: 0,0:06:43.34,0:06:47.34,Default,,0000,0000,0000,,Phase plane and then phase space is \None of the key geometric constructions,
Dialogue: 0,0:06:47.34,0:06:53.87,Default,,0000,0000,0000,,analytical tools used to visualize \Nbehavior of dynamical systems.
Dialogue: 0,0:06:53.87,0:06:59.40,Default,,0000,0000,0000,,So, an important result is that there\Ncan be no chaos, no aperiodic
Dialogue: 0,0:06:59.40,0:07:04.04,Default,,0000,0000,0000,,solutions in 2D differential equations.
Dialogue: 0,0:07:04.04,0:07:08.32,Default,,0000,0000,0000,,So, curves cannot cross in phase space.
Dialogue: 0,0:07:08.32,0:07:12.32,Default,,0000,0000,0000,,The equations are deterministic, and that\Nmeans that every point in space, and
Dialogue: 0,0:07:12.32,0:07:16.32,Default,,0000,0000,0000,,remember this is in phase space, so my\Npoint in space gives the rabbit and fox
Dialogue: 0,0:07:16.32,0:07:24.20,Default,,0000,0000,0000,,population, there's a unique direction\Nassociated with the motion.
Dialogue: 0,0:07:24.20,0:07:28.27,Default,,0000,0000,0000,,DF/DT, DR/DT, that gives a direction.\NIt tells you how the rabbits are
Dialogue: 0,0:07:28.27,0:07:30.53,Default,,0000,0000,0000,,increasing, how the foxes are increasing.
Dialogue: 0,0:07:30.53,0:07:36.19,Default,,0000,0000,0000,,If two phase lines ever cross, like they\Ndo where my knuckles are meeting,
Dialogue: 0,0:07:36.19,0:07:39.71,Default,,0000,0000,0000,,then that would be a non-deterministic\Ndynamical system.
Dialogue: 0,0:07:39.71,0:07:43.71,Default,,0000,0000,0000,,There would be two possible trajectories\Ncoming from one point.
Dialogue: 0,0:07:43.71,0:07:49.15,Default,,0000,0000,0000,,So, the fact that two curves can't cross\Nin these systems limits the behavior.
Dialogue: 0,0:07:49.15,0:07:53.15,Default,,0000,0000,0000,,They sort of literally paint themselves in\Nas they're tracing something out, tracing
Dialogue: 0,0:07:53.15,0:07:57.15,Default,,0000,0000,0000,,a curve out in phase space.
Dialogue: 0,0:07:57.15,0:08:01.15,Default,,0000,0000,0000,,So there can be stable and unstable fixed\Npoints and orbits can tend toward infinity
Dialogue: 0,0:08:01.15,0:08:04.93,Default,,0000,0000,0000,,of course, and there can also be limit\Ncycles attracting cyclic behavior, and we
Dialogue: 0,0:08:04.93,0:08:06.54,Default,,0000,0000,0000,,saw an example of that.
Dialogue: 0,0:08:06.54,0:08:09.45,Default,,0000,0000,0000,,But the main thing is that there can't be \Naperiodic orbits.
Dialogue: 0,0:08:09.45,0:08:12.97,Default,,0000,0000,0000,,And that result is known as the \NPoincaré-Bendixson theorem.
Dialogue: 0,0:08:12.97,0:08:15.81,Default,,0000,0000,0000,,It's about a century old.
Dialogue: 0,0:08:15.81,0:08:19.81,Default,,0000,0000,0000,,And it's not immediately obvious;\Nit takes some proof.
Dialogue: 0,0:08:19.81,0:08:24.91,Default,,0000,0000,0000,,Like I said, that's maybe why it's a \Ntheorem and not just an obvious statement.
Dialogue: 0,0:08:24.91,0:08:29.52,Default,,0000,0000,0000,,One could imagine, and people in the\Nforums have been trying to imagine
Dialogue: 0,0:08:29.52,0:08:39.34,Default,,0000,0000,0000,,space-filling curves that somehow never\Nrepeat but also never leave a bounded area
Dialogue: 0,0:08:39.34,0:08:44.70,Default,,0000,0000,0000,,But the Poincaré-Bendixson theorem says \Nthat those solutions somehow
Dialogue: 0,0:08:44.70,0:08:46.17,Default,,0000,0000,0000,,aren't possible.
Dialogue: 0,0:08:46.17,0:08:48.89,Default,,0000,0000,0000,,So, the main result is that\Ntwo-dimensional differential equations
Dialogue: 0,0:08:48.89,0:08:51.91,Default,,0000,0000,0000,,cannot be chaotic.
Dialogue: 0,0:08:51.91,0:08:54.58,Default,,0000,0000,0000,,That's not the case for three-dimensional\Ndifferential equations, however.
Dialogue: 0,0:08:54.58,0:08:56.92,Default,,0000,0000,0000,,So, here are the Lorenz equations.
Dialogue: 0,0:08:56.92,0:09:00.15,Default,,0000,0000,0000,,Now, again it's a dynamical system, it's\Na rule that tells how something changes
Dialogue: 0,0:09:00.15,0:09:01.25,Default,,0000,0000,0000,,in time.
Dialogue: 0,0:09:01.25,0:09:08.17,Default,,0000,0000,0000,,Here that something is x, y, and z, and\NI forget what parameter values I chose
Dialogue: 0,0:09:08.17,0:09:09.60,Default,,0000,0000,0000,,for sigma, rho, and beta.
Dialogue: 0,0:09:09.60,0:09:12.69,Default,,0000,0000,0000,,And we can get three solutions:\Nx, y, and z.
Dialogue: 0,0:09:12.69,0:09:15.30,Default,,0000,0000,0000,,And these are all curves plotted as a \Nfunction of time.
Dialogue: 0,0:09:15.30,0:09:19.89,Default,,0000,0000,0000,,But we could plot these in phase space,\Nx, y, and z together.
Dialogue: 0,0:09:19.89,0:09:24.12,Default,,0000,0000,0000,,And for that system, if we do that, we get\Nsome complicated structure that
Dialogue: 0,0:09:24.12,0:09:27.43,Default,,0000,0000,0000,,loops around itself and repeats.
Dialogue: 0,0:09:27.43,0:09:30.52,Default,,0000,0000,0000,,It looks like the lines cross, but \Nthey don't.
Dialogue: 0,0:09:30.52,0:09:33.04,Default,,0000,0000,0000,,There's actually a space between them.
Dialogue: 0,0:09:33.04,0:09:36.25,Default,,0000,0000,0000,,It looks like they cross because this is\Na two-dimensional surface trying to
Dialogue: 0,0:09:36.25,0:09:39.70,Default,,0000,0000,0000,,plot something in 3D.
Dialogue: 0,0:09:39.70,0:09:46.59,Default,,0000,0000,0000,,Alright, so just a little bit more about\Nphase space.
Dialogue: 0,0:09:46.59,0:09:49.82,Default,,0000,0000,0000,,Determinism means that curves in phase\Nspace cannot intersect.
Dialogue: 0,0:09:49.82,0:09:53.82,Default,,0000,0000,0000,,But because the space is three-dimensional\Ncurves can go over or under each other.
Dialogue: 0,0:09:53.82,0:09:58.58,Default,,0000,0000,0000,,And that means that there is a lot more \Ninteresting behavior that's possible.
Dialogue: 0,0:09:58.58,0:10:02.58,Default,,0000,0000,0000,,A trajectory can weave around and under\Nand through itself
Dialogue: 0,0:10:02.58,0:10:04.56,Default,,0000,0000,0000,,in some very complicated ways.
Dialogue: 0,0:10:04.56,0:10:08.06,Default,,0000,0000,0000,,What that means, in turn, is that \Nthree-dimensional differential equations
Dialogue: 0,0:10:08.06,0:10:09.37,Default,,0000,0000,0000,,can be chaotic.
Dialogue: 0,0:10:09.37,0:10:16.05,Default,,0000,0000,0000,,You can get bounded, aperiodic orbits,\Nand it has sensitive dependence as well.
Dialogue: 0,0:10:16.05,0:10:19.37,Default,,0000,0000,0000,,And then we saw that chaotic trajectories\Nin phase space are particularly
Dialogue: 0,0:10:19.37,0:10:20.85,Default,,0000,0000,0000,,interesting and fun.
Dialogue: 0,0:10:20.85,0:10:24.25,Default,,0000,0000,0000,,They often get pulled to these things\Ncalled strange attractors.
Dialogue: 0,0:10:24.25,0:10:30.46,Default,,0000,0000,0000,,So here's the Lorenz attractor or the\Nfamous values for the Lorenz equation.
Dialogue: 0,0:10:30.46,0:10:32.74,Default,,0000,0000,0000,,Strange attractors.\NWhat are strange attractors?
Dialogue: 0,0:10:32.74,0:10:36.74,Default,,0000,0000,0000,,Well, they're attractors, and what that \Nmeans is that nearby orbits get
Dialogue: 0,0:10:36.74,0:10:38.02,Default,,0000,0000,0000,,pulled into it.
Dialogue: 0,0:10:38.02,0:10:41.54,Default,,0000,0000,0000,,So, if I have a lot of initial conditions,\Nthey all are going to get
Dialogue: 0,0:10:41.54,0:10:43.18,Default,,0000,0000,0000,,pulled onto that attractor.
Dialogue: 0,0:10:43.18,0:10:47.18,Default,,0000,0000,0000,,So, in that sense, it's stable.\NIf you're on that attractor and somebody
Dialogue: 0,0:10:47.18,0:10:50.34,Default,,0000,0000,0000,,bumps you off a little bit, you'd get\Npulled right back towards it.
Dialogue: 0,0:10:50.34,0:10:52.06,Default,,0000,0000,0000,,That's what it means to be stable.
Dialogue: 0,0:10:52.06,0:10:54.80,Default,,0000,0000,0000,,So, it's a stable structure \Nin phase space.
Dialogue: 0,0:10:54.80,0:10:58.80,Default,,0000,0000,0000,,But the motion on the attractor is not\Nperiodic the way most attractors that
Dialogue: 0,0:10:58.80,0:11:01.03,Default,,0000,0000,0000,,we've seen are, or even fixed points.\N
Dialogue: 0,0:11:01.03,0:11:03.70,Default,,0000,0000,0000,,But the motion on the attractor \Nis chaotic.
Dialogue: 0,0:11:03.70,0:11:07.05,Default,,0000,0000,0000,,So, once you're on the attractor, orbits\Nare aperiodic and have
Dialogue: 0,0:11:07.05,0:11:09.39,Default,,0000,0000,0000,,sensitive dependence on\Ninitial conditions.
Dialogue: 0,0:11:09.39,0:11:12.23,Default,,0000,0000,0000,,So, it's an attracting chaotic attractor.
Dialogue: 0,0:11:16.04,0:11:20.04,Default,,0000,0000,0000,,Then we looked at this a little bit more\Ngeometrically and I argued that the key
Dialogue: 0,0:11:20.04,0:11:23.60,Default,,0000,0000,0000,,ingredients to make a strange attractor\Nor to make chaos of any sort, actually,
Dialogue: 0,0:11:23.60,0:11:25.24,Default,,0000,0000,0000,,is stretching and folding.
Dialogue: 0,0:11:25.24,0:11:28.83,Default,,0000,0000,0000,,So, you need some stretching to pull\Nnearby orbits apart.
Dialogue: 0,0:11:28.83,0:11:32.83,Default,,0000,0000,0000,,The analogy I discussed was\Nkneading dough.
Dialogue: 0,0:11:32.83,0:11:35.61,Default,,0000,0000,0000,,So, when you knead dough, you stretch it.
Dialogue: 0,0:11:35.61,0:11:39.61,Default,,0000,0000,0000,,That pulls things apart, and then you fold\Nit back on itself.
Dialogue: 0,0:11:39.61,0:11:41.47,Default,,0000,0000,0000,,So, the folding keeps orbits bounded.
Dialogue: 0,0:11:41.47,0:11:44.90,Default,,0000,0000,0000,,It takes far apart orbits and moves them\Ncloser together.
Dialogue: 0,0:11:44.90,0:11:47.47,Default,,0000,0000,0000,,But stretching pulls nearby orbits apart,\Nand that's what leads to
Dialogue: 0,0:11:47.47,0:11:50.68,Default,,0000,0000,0000,,the butterfly effect,\Nor sensitive dependence.
Dialogue: 0,0:11:50.68,0:11:55.43,Default,,0000,0000,0000,,Now, stretching and folding, it may be \Nrelatively easy to picture in
Dialogue: 0,0:11:55.43,0:11:59.43,Default,,0000,0000,0000,,three-dimensional space, either a space\Nof actual dough on a bread board
Dialogue: 0,0:11:59.43,0:12:01.14,Default,,0000,0000,0000,,or a phase space.
Dialogue: 0,0:12:01.14,0:12:04.65,Default,,0000,0000,0000,,But it occurs in one-dimensional maps\Nas well, the logistic equation stretches
Dialogue: 0,0:12:04.65,0:12:06.10,Default,,0000,0000,0000,,and folds.
Dialogue: 0,0:12:06.10,0:12:10.10,Default,,0000,0000,0000,,And this can explain how one-dimensional\Nmaps, iterated functions, can capture
Dialogue: 0,0:12:10.10,0:12:14.10,Default,,0000,0000,0000,,some of the features of these\Nhigher dimensional systems.
Dialogue: 0,0:12:14.10,0:12:21.72,Default,,0000,0000,0000,,And it begins to explain, also, how these\Nhigher dimensional systems,
Dialogue: 0,0:12:21.72,0:12:26.73,Default,,0000,0000,0000,,convection rolls, dripping faucets,\Ncan be captured by one-dimensional
Dialogue: 0,0:12:26.73,0:12:31.56,Default,,0000,0000,0000,,functions like the logistic equation and\Nthis universal parameter, 4.669.
Dialogue: 0,0:12:31.56,0:12:36.26,Default,,0000,0000,0000,,So, in any event, stretching and folding \Nare the key ingredients for a chaotic
Dialogue: 0,0:12:36.26,0:12:38.47,Default,,0000,0000,0000,,dynamical system.
Dialogue: 0,0:12:38.47,0:12:43.53,Default,,0000,0000,0000,,So, strange attractors once more. \NThey're these complex structures that
Dialogue: 0,0:12:43.53,0:12:45.78,Default,,0000,0000,0000,,arise from simple dynamical systems.
Dialogue: 0,0:12:45.78,0:12:48.58,Default,,0000,0000,0000,,A reminder that we looked at three\Nexamples: the Hénon map, the Hénon
Dialogue: 0,0:12:48.58,0:12:54.47,Default,,0000,0000,0000,,attractor, which is a two-dimensional\Ndiscrete, iterated function.
Dialogue: 0,0:12:54.47,0:13:00.09,Default,,0000,0000,0000,,And then, two different sets of coupled \Ndifferential equations in three dimensions
Dialogue: 0,0:13:00.09,0:13:04.48,Default,,0000,0000,0000,,the famous Lorenz equations, and also\Nthe slightly less famous but equally
Dialogue: 0,0:13:04.48,0:13:06.88,Default,,0000,0000,0000,,beautiful Rössler equations.
Dialogue: 0,0:13:06.88,0:13:10.64,Default,,0000,0000,0000,,Again, the motion on the attractor is\Nchaotic, but all orbits get pulled
Dialogue: 0,0:13:10.64,0:13:11.62,Default,,0000,0000,0000,,to the attractor.
Dialogue: 0,0:13:11.62,0:13:15.62,Default,,0000,0000,0000,,So, strange attractors combine elements\Nof order and disorder.
Dialogue: 0,0:13:15.62,0:13:18.16,Default,,0000,0000,0000,,That's one of the key themes \Nof the course.
Dialogue: 0,0:13:18.16,0:13:20.88,Default,,0000,0000,0000,,The motion on the attractor is locally\Nunstable.
Dialogue: 0,0:13:20.88,0:13:24.88,Default,,0000,0000,0000,,Nearby orbits are getting pulled apart,\Nbut globally it's stable.
Dialogue: 0,0:13:24.88,0:13:31.88,Default,,0000,0000,0000,,One has these stable structures, the same\NLorenz attractor appears all the time.
Dialogue: 0,0:13:31.88,0:13:35.38,Default,,0000,0000,0000,,If you're on the attractor, you get pushed\Noff it, you get pulled right back in.
Dialogue: 0,0:13:37.82,0:13:41.82,Default,,0000,0000,0000,,Alright, and the last topic we covered in \Nunit 9 was pattern formation.
Dialogue: 0,0:13:41.82,0:13:45.06,Default,,0000,0000,0000,,So, we've seen throughout the course in\Nthe first 8 units that dynamical systems
Dialogue: 0,0:13:45.06,0:13:46.89,Default,,0000,0000,0000,,are capable of chaos.
Dialogue: 0,0:13:46.89,0:13:49.11,Default,,0000,0000,0000,,That was one of the main results.
Dialogue: 0,0:13:49.11,0:13:51.30,Default,,0000,0000,0000,,Unpredictable, aperiodic behavior.
Dialogue: 0,0:13:51.30,0:13:54.25,Default,,0000,0000,0000,,But there's a lot more to dynamical\Nsystems than chaos.
Dialogue: 0,0:13:54.25,0:13:58.25,Default,,0000,0000,0000,,They can produce patterns, structure,\Norganization, complexity, and so on.
Dialogue: 0,0:13:58.25,0:14:01.10,Default,,0000,0000,0000,,And we looked at just one example\Nof a pattern-forming system.
Dialogue: 0,0:14:01.10,0:14:03.28,Default,,0000,0000,0000,,There are many, many ones to choose from.
Dialogue: 0,0:14:03.28,0:14:05.69,Default,,0000,0000,0000,,But we looked at reaction-diffusion\Nsystems.
Dialogue: 0,0:14:07.49,0:14:10.30,Default,,0000,0000,0000,,So there, we have two chemicals that \Nreact and diffuse.
Dialogue: 0,0:14:10.69,0:14:15.74,Default,,0000,0000,0000,,And diffusion, that's just the random\Nspreading out of molecules in space,
Dialogue: 0,0:14:15.74,0:14:19.62,Default,,0000,0000,0000,,diffusion tends to smooth out \Ndifferences, it makes everything
Dialogue: 0,0:14:19.62,0:14:22.32,Default,,0000,0000,0000,,as boring and bland as possible.
Dialogue: 0,0:14:22.32,0:14:26.32,Default,,0000,0000,0000,,But, if we have two different chemicals\Nthat react in a certain way,
Dialogue: 0,0:14:26.32,0:14:29.12,Default,,0000,0000,0000,,it's possible to get stable spatial \Nstructures even in the presence
Dialogue: 0,0:14:29.12,0:14:30.47,Default,,0000,0000,0000,,of diffusion.
Dialogue: 0,0:14:30.47,0:14:33.62,Default,,0000,0000,0000,,Here are these equations-- I described\Nthem in the last unit.
Dialogue: 0,0:14:33.62,0:14:38.05,Default,,0000,0000,0000,,This is deterministic, just like the \Ndynamical systems we've studied before.
Dialogue: 0,0:14:38.05,0:14:42.62,Default,,0000,0000,0000,,And it's spatially extended, because now\NU and V are functions, not just of T, but
Dialogue: 0,0:14:42.62,0:14:44.16,Default,,0000,0000,0000,,of X and Y.
Dialogue: 0,0:14:44.16,0:14:47.39,Default,,0000,0000,0000,,So, these become partial differential\Nequations.
Dialogue: 0,0:14:47.39,0:14:49.87,Default,,0000,0000,0000,,Crucially, the rule is local.
Dialogue: 0,0:14:49.87,0:14:55.07,Default,,0000,0000,0000,,So, the value of U or the value of V, \Nthose are chemical concentrations,
Dialogue: 0,0:14:55.07,0:14:59.82,Default,,0000,0000,0000,,depend on some function of a current\Nvalue at that location, and on this
Dialogue: 0,0:14:59.82,0:15:03.53,Default,,0000,0000,0000,,Laplacian derivative at that location.
Dialogue: 0,0:15:03.53,0:15:10.05,Default,,0000,0000,0000,,So, we have a local rule in that the \Nchemical concentration here doesn't know
Dialogue: 0,0:15:10.05,0:15:13.96,Default,,0000,0000,0000,,directly what the chemical concentration\Nis here; it's just doing it's own thing
Dialogue: 0,0:15:13.96,0:15:15.88,Default,,0000,0000,0000,,at its own local location.
Dialogue: 0,0:15:15.88,0:15:19.90,Default,,0000,0000,0000,,Nevertheless, it produces these \Nlarge-scale structures.
Dialogue: 0,0:15:20.96,0:15:23.68,Default,,0000,0000,0000,,So, just one quick example.
Dialogue: 0,0:15:23.68,0:15:26.47,Default,,0000,0000,0000,,We experimented with reaction-diffusion\Nequations at the
Dialogue: 0,0:15:26.47,0:15:28.83,Default,,0000,0000,0000,,Experimentarium Digitale site.
Dialogue: 0,0:15:28.83,0:15:31.95,Default,,0000,0000,0000,,Here's an example that we saw emerging\Nfrom random initial conditions,
Dialogue: 0,0:15:31.95,0:15:35.17,Default,,0000,0000,0000,,these stable spots appear.
Dialogue: 0,0:15:35.17,0:15:41.83,Default,,0000,0000,0000,,And then we also looked at a video from\NStephen Morris at Toronto where two fluids
Dialogue: 0,0:15:41.83,0:15:46.66,Default,,0000,0000,0000,,are poured into this petri dish, and like\Nmagic, these patterns start to emerge
Dialogue: 0,0:15:46.66,0:15:47.76,Default,,0000,0000,0000,,out of them.
Dialogue: 0,0:15:47.76,0:15:51.44,Default,,0000,0000,0000,,So, Belousov Zhabotinsky has another\Nexample of a reaction-diffusion system.
Dialogue: 0,0:15:53.24,0:15:56.15,Default,,0000,0000,0000,,So, pattern formation is a giant subject.
Dialogue: 0,0:15:56.15,0:15:58.83,Default,,0000,0000,0000,,It could be probably a course in and of\Nitself.
Dialogue: 0,0:15:58.83,0:16:01.70,Default,,0000,0000,0000,,The main point I want to make is that\Nthere's more to dynamical systems
Dialogue: 0,0:16:01.70,0:16:04.96,Default,,0000,0000,0000,,than just chaos or unpredictability\Nor irregularity.
Dialogue: 0,0:16:04.96,0:16:09.21,Default,,0000,0000,0000,,Simple, spatially-extended dynamical\Nsystems with local rules are capable
Dialogue: 0,0:16:09.21,0:16:13.21,Default,,0000,0000,0000,,of producing stable, global patterns and\Nstructures.
Dialogue: 0,0:16:13.21,0:16:16.67,Default,,0000,0000,0000,,So, there's a lot more to the study of\Nchaos than chaos.
Dialogue: 0,0:16:16.67,0:16:20.67,Default,,0000,0000,0000,,Simple dynamical systems can produce\Ncomplexity and all sorts of interesting
Dialogue: 0,0:16:20.67,0:16:22.98,Default,,0000,0000,0000,,emergent structures and phenomena.