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Okay, everybody. Good morning. I hope those of you taking the course officially will turn in at the end of the lecture if you haven't already.
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The assignment. And as it says here, the next one will be handed out on Thursday and do two weeks from today.
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So to give you a little bit of the big picture, this is week two and we're talking about odds.
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Today will be about chaos and related things and then Thursday about boundary value problems as opposed to initial value problems.
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Then on next week, we're going to move towards Pdes partial differential equations.
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But today is a lot of fun with this beautiful topic of chaos that you've all heard of and you all know something about.
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And it's very hard not to enjoy this subject.
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I wanted to begin with some discussion of planetary motions, not just planets, but asteroids and comets and so on.
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And only two days ago, we had the eclipse of the moon in English weather.
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One never sees these things. Really. Did anyone actually see an eclipse?
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There were rumours that in Liverpool you could see something, but not here anyway.
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Let's start by showing a demo you just did.
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Those of you taking the course tested the assignment involving a chaotic orbit of a very artificial sort.
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So we had three fixed suns or planets and then another body orbiting around them in a chaotic fashion.
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Let me show you another example that's a favourite of mine. Which I got originally from James Binney in theoretical physics.
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And this is can be found in a code called M 29 planets.
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So if I run that code. You'll see that.
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It shows you three planets coloured green, red and yellow, which of course are just point masses.
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And the thing that's memorable about them is that they're starting motionless in the configuration of a three,
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four or five triangle, so that you can always remember it gets the right angle triangle three, four or five.
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And nevertheless, what happens is interesting. So let me show you.
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This is a computation, of course, in MATLAB.
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These are random dots in the back. That's really the genius of this code.
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The whole cosmos is produced with a random number generator.
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You can see the Times 72, 78, and now at time 86.
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See what happens. So up to time, 86, it looks like chaos.
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Well, it is chaos in some sense. But then at time, 86, something surprising happens, which could be called ionisation or self ionisation,
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where two of the particles go off in one direction to infinity and the other in the other direction.
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And if you think about it, with just two planets, that couldn't happen, but with three, that is possible.
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You can satisfy conservation of momentum and energy.
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Let's just run that again to see how it looks.
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So one more time. So you see, it looks like standard chaotic motion.
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There's the time up there. So at 86. The ionisation takes place.
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I think that's beautiful. It's one of the remarkable things about chaos is that.
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Without computers, of course, it was very hard to see chaos.
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And for hundreds of years people did odds and the whole subject of chaos wasn't really there.
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Now, PoincarĂ©, the great French mathematician, did sort of figure it out.
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But without computers, it just wasn't so conspicuous.
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As soon as you get computers, it's everywhere.
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It's very hard to write down a system of dynamical equations with more than a few variables, which doesn't give chaotic solutions.
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As an illustration, I'm going to run the same code again, but perturbing the initial condition.
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So this one is called M 29 B.
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And it's exactly the same code, except that the particle that begins at position three, I guess it's the vertical one now begins at position 3.01.
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So it's a perturbation by a 30th of a percent.
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And one of the initial conditions. Why am I not seeing anything?
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Interesting. That's better.
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Okay, so it looks the same because you can't tell that one of them has been perturbed
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slightly and for a while the motions are going to be essentially the same.
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Now, of course, you don't remember the details, but if we get to time 86 for this one.
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You'll notice that's nearly ionised, but not quite. And nothing special happens.
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This code keeps going to run to time. 2200.
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There's still another planet over there. They're going to come back together slowly.
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Yeah. There they are. Does this one?
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Eventually, I and I probably I don't know, maybe there's even a theorem that with a set of initial conditions of a dense set,
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maybe they'd always I and I says, I don't know that you can't run a thing like this without having questions.
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Here's a question. Notice I cooked it up so that the three planets come pretty close together at time.
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200. Suppose one wanted that to happen.
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Exactly. That's probably possible.
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At least at a time. Near 200. It'd be a crazy thing to say, but, you know, maybe we want an actual impact at a time.
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200 or time, close to 200. Could one find initial conditions that do that?
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Who knows? So it's easy to have fun with planetary orbits.
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I want to say a few things about that subject. So first of all, regarding numerical methods, I'm not going to go into what people actually do,
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but some of the good codes for this sort of thing are often built for second order oddities rather than first order.
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If you really care about accuracy,
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you can design finite differences that democratise the second derivative rather than a system of equations with the first derivative.
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And a famous name in this field are called Sturmer formulas or Sturmer valet formulas, and these work with second order equations.
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Another thing I want to mention is so called geometric integration.
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This is a hot topic in the academic and practical side of numerical solution of odds.
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A problem is physical, as this has all kinds of interesting things going on,
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like conservation of momentum and energy and other conserved quantities sometimes.
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And there's a field of geometric integrators which make special formulas that conserve appropriate things.
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Another word that comes up in this business is symplectic integrators and so too a dynamic system.
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Problems like this have a lot of mathematical symplectic structure that one can take advantage of, and there is a lot of work on that.
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Let me mention in particular, if you're interested in this, there's an outstanding book by, well, two names you've already seen.
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I've passed around a book by Hira and Varner, but the book on this particular subject is by three of them, also Luby.
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So high were Varner and Lubitsch. Three Austrians have a great book on numerical methods for geometric integration.
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But I wanted to say a bit more about planets. So I got curious years ago when they tell you there's going to be an eclipse.
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Where does that information come from? So of course, you'll get it from the newspaper, the television or the web or something.
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Where do they get it from? Well, usually they get it from some national organisation, such as?
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I've listed a few the Admiralty office, the Royal Greenwich Observatory, the British Astronomical Society,
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NASA Goddard Space Centre, the U.S. Naval Observatory, the in France, the Bureau de Longitude.
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The Indian Department of Meteorology. The Astronomy Astronomical Division of the Department of Hydrography in the University of Tokyo.
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Places like this give data about eclipses. Now, where do they get that data?
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It turns out almost all of them get it from Southern California.
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It's all JPL, the Jet Propulsion Lab. So that is the answer to the question.
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All these calculations, mostly more than 90% of them come from the SS.
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D is the Solar System Dynamics Group. And this is at the Jet Propulsion Laboratory, JPL, in Pasadena, California.
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That's where most of the calculations are done.
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I believe the French actually do it themselves, but all the people who are not French rely on Southern California, so far as I know.
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The government shutdown has not affected this.
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I guess the Jet Propulsion Laboratory is clever enough to insulate its fundings from the whims of the President.
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But I copied for you there Home Page. It's not a very beautiful and exciting page to look at, but what it links to is really quite incredible.
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Through here you can get all the details. So just just look at this.
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These numbers. Their standard system tracks 791,373.
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Asteroids, 3557 comets, 190 planetary satellites, eight planets.
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So that answers our question about Pluto, the Sun, various spacecraft and so on.
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It's an amazing system using these stern more fairly formulas.
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I believe it does this to remarkable accuracy.
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They can make predictions thousands of years in the future and thousands of years in the past in order to track historical records.
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The accuracies depend on how far away you are in time.
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If you're within a few decades, you get unbelievable accuracies in centimetres or metres,
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depending on exactly what you're measuring and even a thousands of years away.
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You get good accuracy. It's an amazingly mature subject, which for my work, in my opinion,
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is really the great example of very high end calculations conducted by the human race, basically.
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Very exciting stuff. Now.
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What else that I want to mention about that. Oh, I believe the actual body integrators they use have all sorts of add captivity built into them.
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They have variable orders, variable step sizes. It's it's a wonderfully mature technology.
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Have any of you connected with this stuff to any of your work in such areas?
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Uh huh. So what department are you in? Uh huh.
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Yeah. Okay. And by the way, I think a lot of the bodies are essentially point masses.
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That is good enough for many purposes. The Earth isn't the point mass, but further off planets often are.
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Relativistic effects are important. Most of the bodies have that included in them.
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So that's a great thing to be aware. So let's say a word about chaos.
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And this is a huge subject.
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Let me in particular include in my heading the notion of leaping off exponents.
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So as I say, chaos has always been there but never really got focussed on until computers came along.
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Really in the seventies I guess is when it hit the headlines.
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The central question. As people would regard it nowadays.
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Four possibly chaotic systems would be this.
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If the initial data are perturbed.
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And it even makes sense mathematically to talk about infinitesimal perturbations.
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Of course, in practice, no perturbation will be infinitesimal.
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But mathematically, you can imagine an infinitesimal perturbation.
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And you can ask how fast? Does it grow with time?
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And the advantage of infinitesimal perturbations is that you can let time go as long as you want, and they're still infinitesimal.
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The big distinction is between systems where perturbations grow algebraically and perturbations where systems where they grow exponentially.
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So algebraic growth means of time to some power.
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Exponential growth. Means O of E to some number like lambda T, where lambda is bigger than zero,
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or the real part of lambda is bigger than zero, depending exactly what you're measuring.
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Now a chaotic system needs to have this second property. That's not all it needs.
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For example. Here's an ode to EU Prime equals U.
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The solution is U of T equals either the t.
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That solution grows exponentially. And moreover, if you perturb the initial condition, that perturbation will grow exponentially.
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But of course, that's not interesting. The interesting cases are where you get exponential growth with a global bounded ness of some kind.
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So this is the situation where chaos turns up.
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You have exponential growth of perturbations, but still you have in some sense some kind of approximation to global bounded ness of orbits.
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Now, I'm not trying to give a precise definition.
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And in fact. Although chaos does have precise definitions,
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this is not an area where people have settled very happily on one particular definition that everybody uses.
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It's a bit complicated, but you know it when you see it, as they say.
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Probably most of you know about STROGATZ, his book.
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So here's one of the editions of it.
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Steve STROGATZ from Cornell was an early person writing a text book, and it's just a fantastic book that everybody loves.
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So, Steve STROGATZ, if you want to know about chaos, that's the place to start.
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I'm curious. Raise your hand if you were aware of STROGATZ this book.
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Yeah, that's impressive. Raise your hand if you've looked at. Have his videos online about chaos.
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Okay. He's also a person in the news.
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He writes columns for the New York Times and for other magazines and so on.
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So he's one of the well known applied mathematicians on the planet.
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Just to say a word about the history. So I mentioned John Kerry, who was very important in the pre-history, if you like.
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And that's going back to the nine, you know, 1900 ish.
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He was concerned with the Three-body problem and the solar system and so on,
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and he realised that effectively chaotic things could happen, though he didn't coin that term.
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Some other key names were in the 1970s.
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There was Roussel and Tompkins and there was Feigenbaum was a key person, so-called universality.
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And then there was Bob May, who became Lord May, who showed chaos in biological systems.
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And there was Jim York from Maryland who invented the term.
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But then the really famous original person was Ed Lorenz.
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So. Ed Lawrence at MIT.
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And was it 1963 wrote the great paper that sort of put chaos on the map, even though the name wasn't there yet.
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He died about ten years ago. He lived to be 90 years old.
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I checked this morning on the Web, you get 453 million hits on the word chaos.
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They're not all mathematical chaos, but many of them are. And if you get a glance at the others, I've given you the first page of his paper.
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This is really a great paper. It's so readable.
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If you look at it, you just feel inspired for how much he saw using the primitive computers of those days.
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It's it's really great.
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He was a a quiet kind of he was kind of an on the spectrum, brilliant genius who didn't say much but knew he was on to something very real here.
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So Ed Lorenz. Let me give some examples.
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As I say, it's pretty hard not to find chaos if you look at a big dynamical system.
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So one field in which people look for chaos is what they call billiards.
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Billiards means bouncing particles. In the simplest case, you have a billiard table and you say,
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What about the orbits of a ball bouncing around that table if the billiards are on a rectangular table?
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That's not chaotic. The perturbations grow only algebraically.
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On the other hand, if you have billiards on a curved table, well, it's not always chaotic, but it's depending on precisely the shape it usually is.
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And there's a whole business of quantum chaos in which people study chaotic and non chaotic orbits of particles moving around in point particles,
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which relates to eigen functions of the in the plus operator in quantum mechanics.
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It's a highly developed field. So billiards is one example.
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Here's another example if you have any linear equation.
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No matter how many variables, it's not chaotic.
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It can very easily have exponential growth of perturbations, but that would be in the context of actual solutions going off to infinity.
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You don't have the bounded mass needed for things to be really complicated.
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Another thing that is never chaotic. Is any one variable or two variable system.
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First order autonomous ODI.
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So such a problem cannot be chaotic.
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An example would be the Vanderpoel equation that we've looked at. And the reason is.
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If you have, let's say, two variables, your trajectories are trapped in a plane.
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They're completely described by a phase plane. So.
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Suppose you're in a plane. There are theorems about this.
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Of course. It's impossible to get tangled up in a plane because you can't cross a trajectory.
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An autonomous system is completely determined by its value at a point.
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So you could have trajectories that look like this, but notice how constrained they are.
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They can't really. Since they can't cross, they can't really get tangled.
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So there's the hand-waving argument that two dimensional systems cannot be chaotic.
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But as soon as you have three dimensions, then chaos is very common.
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Now. What about the weather on planet Earth? Which was.
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Lawrence's motivation. Well, surely that is chaotic.
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It's a bit hard to say because, of course, that's not a well-defined problem.
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It's an infinitely complicated system. So, properly speaking, chaos applies or doesn't apply to a precise set of equations,
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but many reasonable sets of equations that model Earth are indeed chaotic.
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A famous one is turbulent flow in a pipe.
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Now it's obvious that, loosely speaking, that's chaotic.
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It's very hard to predict crazy things happen.
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But it's an example of how challenging it can sometimes be to connect mathematics and physics that the rigorous study of this is a bit controversial.
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And there are those who have argued that. What looks like chaos in a turbulent flow.
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Is actually, in theory, always transient. So it will look chaotic for a time like our planets that eventually ionise,
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but then eventually it has to become a smooth flow again eventually can mean on a doubly exponential timescale.
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So it's this is spectacularly academic, that discussion.
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Obviously in practice it's chaotic, but it's not completely clear that in an infinite circular pipe modelled by the navier-stokes equations,
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technically speaking, one has chaos. Another example.
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Now. I guess that's all I have on my list. But let's say a little bit about planets.
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About the solar system. So the solar system has a lot of particles in it.
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So you would think, given that even three particles, it would seem, can be chaotic.
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Obviously, the solar system is going to be chaotic. The reason it's not so simple is that the sun is so big and heavy.
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So there are a lot of epsilon in the solar system compared with the big sun in the middle.
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So there's actually some controversy here, too.
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But I think most people now believe it is chaotic.
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So it would seem that with timescales on the order of tens of millions of years or more.
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Now, what does it mean to be chaotic?
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Basically, the sort of question that arises is might the solar system eject one of the planets out to infinity in principle?
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And I think many people now think that perhaps yes, the answer in principle is yes, though it won't happen in a hurry.
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Let me mention some key names here. Just two pairs of names in the 1990s.
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Sussman and Widom at MIT. Didn't major calculations which sort of put this problem back on the map after Poincare.
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And then more recently in France, the big names, a big name really is Lashkar.
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And another key person is Gastineau. Okay.
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I guess that's all I'll say. Well, I'll say one more thing.
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Chaos is a phenomenon, but also it's something of interest in engineering and people.
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It's such a huge thing. People have done everything with it.
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For example, if a system is nearly random, then it should be that if you want to achieve a particular outcome,
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some set of initial conditions in principle will do that.
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Like I said, might there be initial conditions that at times 200 made those planets collide?
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And so this would be the general subject that people might call control of chaos,
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where you're using the exponential divergence to achieve interesting things.
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It's been used in cryptography, for example. You can interplay between the deterministic and the seemingly random aspects to encode messages.
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So it was a huge subject. Now I want to say something a little more precise.
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So let's talk about one of the favourite examples, which is the Lorenz equations, and I'll call that the next example.
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These equations are often the first place that people see chaos.
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But for my money, planets orbiting around each other are really the most obvious example, as in what we just looked at.
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But let's call this section 4.10.
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So these equations go back to 1963.
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Now you know that to get chaos, you need at least three variables and you need to be non-linear.
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And sure enough, there are three variables and it's non-linear.
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So in some sense this is as simple as it can get and still be chaotic.
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And let me write down the most famous version of these equations.
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X prime equals ten Y minus ten x.
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That's a linear equation for how X evolves.
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Why Prime equals 28 x.
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Minus Y, minus x, z.
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So that's a nonlinear equation for how Y evolves. And Z prime is x.
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Times Y minus eight thirds. Time Z.
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So that's another non-linear equation.
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Notice the nonlinear nonlinear sureties are as simple as you could ask for just quadratic products of two terms.
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The precise numbers can be changed. It doesn't have to be exactly 28 and 8/3 and ten.
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But these particular choices have become so standard that 99% of discussions of the Lorenz equations use these numbers.
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So if you start analysing this, what you find is that one solution?
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Is X equals Y equals z equals zero.
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Plug that in. And you see, of course, the right hand sides are all zero.
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Therefore, the equation is satisfied.
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So this would be called a fixed point of this dynamical system.
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A fixed point is a point which doesn't vary with time steady state.
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It's an unstable fixed point, but it is a fixed point. There are two more fixed points.
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And they are at x equals y equals plus or minus six square root of two.
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And Z equals 27. So if you analyse those, plug these into the equation, you'll find that again, the derivatives are all zeros.
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So it's a fixed point. It's unstable, meaning that small perturbations will grow away from those fixed points.
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So when something's unstable, in a physical sense, you would expect not to observe it in the real world.
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Now what happens is that when you compute most solutions to these equations, they move around in a seemingly random fashion.
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Lorenz claimed that in 1963, many people thought he was wrong.
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They argued that it was errors in the computer. You know, this is 1963 computers.
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Just imagine how slow and moreover, the floating point arithmetic in those days was not as established as now.
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Every machine was different. People didn't understand it so well.
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It was hard for him to persuade people that it was true. But he was very careful and he did persuade people.
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It took 36 years. For a proof.
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So there's a famous guy called Warwick Tucker, who I guess is the head of the math department at the University of Uppsala.
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And he wrote a paper called The Lorenz Attractor Exists.
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So this is the gulf between pure and applied mathematics.
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With chaotic systems, it can be very hard to prove things. But he really did prove rigorously, though, using computer assistance of a certain sort.
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He proved that there is a strange attractor, as they call it, for the Lorenz equation.
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And this was in a famous French journal in 1999.
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What's the. Layup on off exponent.
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I didn't define it, did I. But the layup on off exponent which is in my heading.
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Is Lambda. So it's, it's the lambda when we get E to the lambda t behaviour and more precisely it's the maximum such,
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so it's the maximum exponential rate at which perturbations can grow infinitesimal perturbations.
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These things are hard to calculate. I think for the Lorenz system it's thought to be about 0.9057.
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So about one. So perturbations grow on average at a rate of about E to the T.
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The strange attractor, which I haven't tried to define and I won't, is a mathematically precise sense of what the crazy orbits look like.
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It's a set in three dimensions.
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The reason they're three dimensions is that we have X and Y and Z.
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And so the strange attractor is a three dimensional object.
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But one of the wonderful things.
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About chaos is that these are geometric objects, although they sit in three dimensions, may have a lower dimensionality in a precise sense.
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And I believe that the dimension of the strange attractor is known to quite a few digits.
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It's something like 2.0.
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6 to 7 1600.
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I'm going to plot it in a second. So what that means is that the chaotic orbits of the Lorenz equations.
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After an initial period, essentially lie on a object which is nearly two dimensional, but slightly more so.
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It's a little thicker than a two dimensional manifold.
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So let's play with that now. Okay.
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There are lots of ways to play with the Lorenz equations. Everybody's got a method.
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So let's start with Matt Labs method. If I type Lorenz in MATLAB, I there's a built in demo.
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So let's see what that does. This has been there for ages.
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I'll press start and you see it just.
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Uses Odie something or other. I don't know which one to compute a trajectory.
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And you can see the this is the strange attractor emerging as you have this chaotic motion.
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You see it sort of looks two dimensional. It's got these two butterfly wings, if you like, but they're not quite flat.
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If you rotated around,
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you'd see there's a little bit of thickness there and there is a precise way that mathematicians measure these non integer dimensions.
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Any comments or questions. Stop.
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Let's do another demo. Let's try and check.
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Fine. I'll use a cheap, gooey. So the graphical user interface to the O.D initial value code.
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So there we are, one of the built in demos is.
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The Lorenz equations. This is a much slower way to solve the problem.
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Telephone is never the fastest, so it's a coupled initial value problem, a system.
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And I'll click on Lorenz Equations. And then if I type solve, you can see it's only going up to time 15.
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So this is a very short trajectory. And then in the end it's plotting the three components U, V, W, X, Y, and Z.
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And you can see this familiar random looking behaviour.
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One of the nice things about a system like this is that you can play around and change things very easily.
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So what happens if you change some of those coefficient ten or 28 or 8/3?
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I'll just make one change that I happen to know does something interesting.
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I'll change 28 to 22. So there you can see that for a while it looks sort of similar, but it's obviously then settling down to a periodic orbit.
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So that's not a chaotic system, evidently. Let me also draw your attention to the book Exploring Odes, which I mentioned.
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So if you look in there, of course, there's a chapter on Chaos, chapter 13, I think.
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But then if you go to this appendix, I like so much appendix B with the more examples.
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A bunch of those examples are chaotic systems, so I think it starts with 64.
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So in fact, when we were writing the book. I asked myself whether.
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Please. Can you tell if this is working? Yeah. I asked myself whether you might have a scalar, chaotic system.
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Of course. In principle, it must be possible, but it has to be at least a third order because you need three variables y y prime and wide.
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So sure enough, if you look on the web for scalar chaotic systems, you find one.
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And this is a simplified version of that.
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Notice is a third derivative here and we've cooked it up so that you're getting chaotic behaviour in this scalar ODI.
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If you look at the other examples, I won't linger.
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But so there you have a forced nonlinear pendulum which appears in some books.
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Here you have the Roessler equations. This particular orbit is periodic, but you can see a kind of a period doubling phenomenon.
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With some parameter it would have been simply periodic. Now you can see the two different amplitudes reflect a doubling of the period.
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This is the universal so-called root chaos that Feigenbaum made famous.
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Here's a three body problem, just as we've been looking at.
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This one is not interesting. The initial conditions don't do anything remarkable.
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This one, on the other hand, is interesting, chaotic.
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Here's a double pendulum. That's a nice example that you often see in people's offices because you can build one physically.
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Here's the so-called hand on health equations.
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And I guess that's it for the chaos. Now.
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I thought I had an example that showed the growth of perturbations.
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Was that the very first one? Did I fail to? Where am I?
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Here. Let me remind myself if I do em again.
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No, that's not the one I want. Uh, I don't think it is.
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Well. I'll run that while I figure out what I'm looking for.
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No, I guess it's just in the next one I'm going to show you. I thought I had one already.
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Okay. So let me finish up by discussing another famous example of chaos in the billiard department.
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So if you think of billiards, the physics is so simple.
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The particle is always going at a fixed speed. So how many variables are there?
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Well, at any time it has a position, and that's two variables.
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And it has a direction which is a third variable. So billiards has just the right count to be interesting from a chaos point of view.
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And the example I want to mention is called the senior billiard.
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So this is our last example. Seni is a famous mathematician and physicist billiard.
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And I'm also going to mention the so-called Siam 100 degree challenge.
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Which I think came up earlier when we talked about optimisation in two dimensions.
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So if you look at the handout, you'll see that the.
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The reverse side of Lorenz's paper is this thing that I published back in 2002 where I asked people around the world to solve ten problems,
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each to ten digits of accuracy. So these are the ten problems.
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Nine of the problems are not chaotic. And what that means in practice is that you can get ten digits of accuracy on an ordinary 16 digit computer.
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However, one of the problems is chaotic, and that's number two.
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So if you look at problem number two.
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You see, the way it's described there is that you have an infinite set of circles in the plane.
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And you have a particle moving around and a particular.
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Orbit is specified doing things like this. Well, that's a chaotic system.
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It's a billiard problem in which the boundary is not just straight lines, but curves.
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And it can be proved that that's chaotic. In order to prove that, you don't really need to think of the infinite domain like this.
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What Seni did was realise that the simplest version of this would be.
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That kind of domain. Because if a particle.
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Keeps going. Well, that's equivalent essentially to being reflected.
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So it turns out. The compliment of an infinite set of circles is equivalent to the interior of this simply connected so-called seni billiard.
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So that is his billiard. And he proved that the motion is chaotic.
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And I want to show you that in action. So this is a code that I actually haven't printed up, though.
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I'll put it on the web called M 30 Circles.
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So suppose I say am 30 circles. And.
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It asks how much of a perturbation I want. So I'll start with a perturbation of ten to the minus.
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Ten and you'll see as we run it what that means.
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I'll tell you before we run it,
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we're going to track the 100 digit challenge problem with the specified initial condition and then also with a perturbed initial condition.
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And we're going to compare the two orbits. So.
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You can see that's actually two particles bouncing around.
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And then finally, after a while, they become distinguishable.
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The red one has gone off. Who knows where? And the black one is still there.
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So. Let's press. No, I'm going to run it once more before pressing.
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Enter. Okay. Let's try a different perturbation.
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I'll try ten to the -12 now.
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10 to -8. We keep losing the particle.
386
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I want to find one that where we keep the red particle. So I try ten to the minus nine.
387
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Must be possible a. It keeps.
388
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Okay. That's a little better.
389
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Okay, good. I like that one more. So you can see that for a certain time.
390
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Although the trajectories are diverging exponentially, they're so close that you don't see that in the in the picture.
391
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But then eventually they have diverged so much that.
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They're no longer correlated in any way. Now what happens at that point is that.
393
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The exponential divergence ceases. Everything is now algebraic.
394
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Nothing is growing exponentially in this picture. Apart from those initial perturbations.
395
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And that's when I press return the second plot that comes up.
396
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So this plot shows the distance between the red and the black dots as a function of time.
397
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And we started with perturbation ten to the minus nine. And you see a pretty systematic exponential growth.
398
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The slope of that curve is the the up and off exponent. But then once it gets to scale one, no further exponential growth can possibly happen.
399
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So you see, mathematically, if we started with an infinitesimal perturbation, this behaviour in principle would go forever.
400
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Let's do it one more time with a very small perturbation. Suppose I say.
401
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And 30. I'll say one E -15 and see what happens.
402
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Okay. So there you can see the exponential growth is just barely beginning to level off at the end.
403
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I think now we still have three more minutes, so I want to go back and play some more with the Lorenz equations unless somebody has a question.
404
00:46:57,260 --> 00:47:01,200
Yes. Is there a formal way of. Assessing.
405
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Like when faced with the. To assess whether it is stable or not, whether we sort of just sort of just playing with it.
406
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I think the essential answer is, no, there isn't. You just play with it.
407
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And of course, play is a loaded word, but you do very serious computing.
408
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But no. And this is why, for example, the solar system is controversial.
409
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It's very hard to assess these things and small changes in parameters can change the answer.
410
00:47:31,910 --> 00:47:35,930
And there's a lot known about that. This famous Feigenbaum period doubling.
411
00:47:35,930 --> 00:47:42,890
Yeah, there's an infinite cascade as you've change a parameter and the different regimes get exponentially closer and closer together.
412
00:47:43,610 --> 00:47:46,370
Although, you know the general structure of such things,
413
00:47:47,150 --> 00:47:53,540
there's no way of knowing a priori whether a particular parameter choice will be a chaotic one or not, I think.
414
00:47:55,430 --> 00:47:59,380
Other questions. I guess I thought it would be fun.
415
00:47:59,470 --> 00:48:03,310
Yes. Another question. The economics.
416
00:48:03,310 --> 00:48:07,960
But it depended on the size of your tax evasion. In principle, no.
417
00:48:07,990 --> 00:48:09,910
So the rigorous definition,
418
00:48:09,910 --> 00:48:18,160
you have to maximise overall infinitesimal perturbations because some of them are maybe going to be in lucky directions that don't excite the growth.
419
00:48:18,580 --> 00:48:22,180
But loosely speaking, it doesn't depend.
420
00:48:25,750 --> 00:48:30,190
I wanted to try, I guess, changing the parameters a couple more times.
421
00:48:30,200 --> 00:48:33,730
So we'll go back to our Lorenz demo.
422
00:48:34,840 --> 00:48:44,850
This was coupled initial value problems. So we know that with ten, 28 and 8/3, we get chaos.
423
00:48:45,180 --> 00:48:48,239
Now, if anyone has a particular choice, I'm happy to listen.
424
00:48:48,240 --> 00:48:53,010
But failing that, let's just try a few things at random. What if we change 28 to 38?
425
00:48:55,000 --> 00:49:01,390
That looks chaotic. Of course, we don't know. What if we changed 22 to 24?
426
00:49:01,420 --> 00:49:04,629
I think I did try this one. Now.
427
00:49:04,630 --> 00:49:08,340
Is that chaotic or periodic? Well, let's run it on a longer interval.
428
00:49:08,350 --> 00:49:16,940
Let's take an interval of 30. It's it seems maybe to be settling down to something.
429
00:49:17,120 --> 00:49:20,510
Let's try an interval of 60. Oops.
430
00:49:25,540 --> 00:49:29,500
Maybe it's not. So that probably is chaotic. But of course, this is no proof.
431
00:49:30,100 --> 00:49:36,310
Let's go back to 28 and change one of the other numbers. So what if we change 10 to 9?
432
00:49:40,330 --> 00:49:49,000
That looks chaotic. What if we changed the five? That's not only periodic but very boring.
433
00:49:49,010 --> 00:49:56,120
It's obviously settled down. So of course there will be some critical number in between at which the behaviour changes.
434
00:49:56,150 --> 00:49:59,720
Six seems to be high enough. What about 5.5?
435
00:50:00,890 --> 00:50:05,600
You can see how addictive this sort of thing is. What about okay, we're going to nail it, right?
436
00:50:05,600 --> 00:50:09,980
5.25. We're doing by six in here now.
437
00:50:10,130 --> 00:50:14,260
5.1, two, five. Ah.
438
00:50:14,540 --> 00:50:18,200
5.1. Oh, I got the wrong way.
439
00:50:18,210 --> 00:50:23,270
Sorry. 5.15. Now 5.175.
440
00:50:25,490 --> 00:50:38,540
5.2. So you see, it's absolutely fascinating and infinitesimal changes can change in theory or whole dynamics.
441
00:50:38,810 --> 00:50:43,550
Okay. Boundary value problems on Thursday. Remember to turn in your assignments, please.