1 00:00:00,330 --> 00:00:08,720 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. 2 00:00:08,730 --> 00:00:14,670 The assignment. And as it says here, the next one will be handed out on Thursday and do two weeks from today. 3 00:00:15,510 --> 00:00:20,790 So to give you a little bit of the big picture, this is week two and we're talking about odds. 4 00:00:21,120 --> 00:00:28,380 Today will be about chaos and related things and then Thursday about boundary value problems as opposed to initial value problems. 5 00:00:29,310 --> 00:00:33,960 Then on next week, we're going to move towards Pdes partial differential equations. 6 00:00:34,200 --> 00:00:39,839 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. 7 00:00:39,840 --> 00:00:43,020 And it's very hard not to enjoy this subject. 8 00:00:43,830 --> 00:00:51,510 I wanted to begin with some discussion of planetary motions, not just planets, but asteroids and comets and so on. 9 00:00:51,900 --> 00:00:57,090 And only two days ago, we had the eclipse of the moon in English weather. 10 00:00:57,120 --> 00:01:00,930 One never sees these things. Really. Did anyone actually see an eclipse? 11 00:01:01,830 --> 00:01:06,600 There were rumours that in Liverpool you could see something, but not here anyway. 12 00:01:08,910 --> 00:01:13,270 Let's start by showing a demo you just did. 13 00:01:13,290 --> 00:01:20,410 Those of you taking the course tested the assignment involving a chaotic orbit of a very artificial sort. 14 00:01:20,430 --> 00:01:27,030 So we had three fixed suns or planets and then another body orbiting around them in a chaotic fashion. 15 00:01:27,780 --> 00:01:36,920 Let me show you another example that's a favourite of mine. Which I got originally from James Binney in theoretical physics. 16 00:01:38,330 --> 00:01:43,460 And this is can be found in a code called M 29 planets. 17 00:01:44,060 --> 00:01:50,590 So if I run that code. You'll see that. 18 00:01:52,010 --> 00:01:59,690 It shows you three planets coloured green, red and yellow, which of course are just point masses. 19 00:01:59,990 --> 00:02:05,899 And the thing that's memorable about them is that they're starting motionless in the configuration of a three, 20 00:02:05,900 --> 00:02:11,780 four or five triangle, so that you can always remember it gets the right angle triangle three, four or five. 21 00:02:12,050 --> 00:02:16,280 And nevertheless, what happens is interesting. So let me show you. 22 00:02:16,280 --> 00:02:20,120 This is a computation, of course, in MATLAB. 23 00:02:23,190 --> 00:02:26,540 These are random dots in the back. That's really the genius of this code. 24 00:02:26,550 --> 00:02:30,840 The whole cosmos is produced with a random number generator. 25 00:02:31,590 --> 00:02:36,840 You can see the Times 72, 78, and now at time 86. 26 00:02:38,280 --> 00:02:42,100 See what happens. So up to time, 86, it looks like chaos. 27 00:02:42,120 --> 00:02:50,880 Well, it is chaos in some sense. But then at time, 86, something surprising happens, which could be called ionisation or self ionisation, 28 00:02:51,090 --> 00:02:56,790 where two of the particles go off in one direction to infinity and the other in the other direction. 29 00:02:57,030 --> 00:03:02,730 And if you think about it, with just two planets, that couldn't happen, but with three, that is possible. 30 00:03:02,880 --> 00:03:05,940 You can satisfy conservation of momentum and energy. 31 00:03:06,180 --> 00:03:09,900 Let's just run that again to see how it looks. 32 00:03:10,410 --> 00:03:18,510 So one more time. So you see, it looks like standard chaotic motion. 33 00:03:27,020 --> 00:03:32,739 There's the time up there. So at 86. The ionisation takes place. 34 00:03:32,740 --> 00:03:38,740 I think that's beautiful. It's one of the remarkable things about chaos is that. 35 00:03:40,190 --> 00:03:43,849 Without computers, of course, it was very hard to see chaos. 36 00:03:43,850 --> 00:03:49,310 And for hundreds of years people did odds and the whole subject of chaos wasn't really there. 37 00:03:49,340 --> 00:03:53,150 Now, Poincaré, the great French mathematician, did sort of figure it out. 38 00:03:53,360 --> 00:03:56,870 But without computers, it just wasn't so conspicuous. 39 00:03:57,050 --> 00:03:59,320 As soon as you get computers, it's everywhere. 40 00:03:59,330 --> 00:04:06,710 It's very hard to write down a system of dynamical equations with more than a few variables, which doesn't give chaotic solutions. 41 00:04:07,520 --> 00:04:13,940 As an illustration, I'm going to run the same code again, but perturbing the initial condition. 42 00:04:14,300 --> 00:04:17,480 So this one is called M 29 B. 43 00:04:19,900 --> 00:04:31,360 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. 44 00:04:31,780 --> 00:04:35,139 So it's a perturbation by a 30th of a percent. 45 00:04:35,140 --> 00:04:41,750 And one of the initial conditions. Why am I not seeing anything? 46 00:04:43,130 --> 00:04:48,640 Interesting. That's better. 47 00:04:48,880 --> 00:04:52,990 Okay, so it looks the same because you can't tell that one of them has been perturbed 48 00:04:52,990 --> 00:04:57,580 slightly and for a while the motions are going to be essentially the same. 49 00:04:59,060 --> 00:05:03,800 Now, of course, you don't remember the details, but if we get to time 86 for this one. 50 00:05:06,060 --> 00:05:10,440 You'll notice that's nearly ionised, but not quite. And nothing special happens. 51 00:05:10,680 --> 00:05:13,980 This code keeps going to run to time. 2200. 52 00:05:17,030 --> 00:05:21,080 There's still another planet over there. They're going to come back together slowly. 53 00:05:22,510 --> 00:05:25,540 Yeah. There they are. Does this one? 54 00:05:25,540 --> 00:05:35,829 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, 55 00:05:35,830 --> 00:05:40,750 maybe they'd always I and I says, I don't know that you can't run a thing like this without having questions. 56 00:05:41,020 --> 00:05:47,040 Here's a question. Notice I cooked it up so that the three planets come pretty close together at time. 57 00:05:47,050 --> 00:05:50,290 200. Suppose one wanted that to happen. 58 00:05:50,290 --> 00:05:53,499 Exactly. That's probably possible. 59 00:05:53,500 --> 00:06:01,209 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. 60 00:06:01,210 --> 00:06:05,980 200 or time, close to 200. Could one find initial conditions that do that? 61 00:06:06,160 --> 00:06:11,580 Who knows? So it's easy to have fun with planetary orbits. 62 00:06:11,730 --> 00:06:24,020 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, 63 00:06:24,200 --> 00:06:36,950 but some of the good codes for this sort of thing are often built for second order oddities rather than first order. 64 00:06:40,220 --> 00:06:42,230 If you really care about accuracy, 65 00:06:42,710 --> 00:06:49,820 you can design finite differences that democratise the second derivative rather than a system of equations with the first derivative. 66 00:06:50,030 --> 00:06:59,600 And a famous name in this field are called Sturmer formulas or Sturmer valet formulas, and these work with second order equations. 67 00:07:00,990 --> 00:07:05,400 Another thing I want to mention is so called geometric integration. 68 00:07:09,020 --> 00:07:16,640 This is a hot topic in the academic and practical side of numerical solution of odds. 69 00:07:18,050 --> 00:07:22,070 A problem is physical, as this has all kinds of interesting things going on, 70 00:07:22,070 --> 00:07:26,390 like conservation of momentum and energy and other conserved quantities sometimes. 71 00:07:26,660 --> 00:07:33,770 And there's a field of geometric integrators which make special formulas that conserve appropriate things. 72 00:07:33,980 --> 00:07:41,630 Another word that comes up in this business is symplectic integrators and so too a dynamic system. 73 00:07:42,110 --> 00:07:49,850 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. 74 00:07:49,910 --> 00:07:57,200 Let me mention in particular, if you're interested in this, there's an outstanding book by, well, two names you've already seen. 75 00:07:57,200 --> 00:08:04,370 I've passed around a book by Hira and Varner, but the book on this particular subject is by three of them, also Luby. 76 00:08:06,520 --> 00:08:13,230 So high were Varner and Lubitsch. Three Austrians have a great book on numerical methods for geometric integration. 77 00:08:20,860 --> 00:08:28,030 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. 78 00:08:28,150 --> 00:08:34,390 Where does that information come from? So of course, you'll get it from the newspaper, the television or the web or something. 79 00:08:34,780 --> 00:08:40,120 Where do they get it from? Well, usually they get it from some national organisation, such as? 80 00:08:40,120 --> 00:08:45,520 I've listed a few the Admiralty office, the Royal Greenwich Observatory, the British Astronomical Society, 81 00:08:45,700 --> 00:08:51,220 NASA Goddard Space Centre, the U.S. Naval Observatory, the in France, the Bureau de Longitude. 82 00:08:51,370 --> 00:08:58,810 The Indian Department of Meteorology. The Astronomy Astronomical Division of the Department of Hydrography in the University of Tokyo. 83 00:08:59,080 --> 00:09:04,090 Places like this give data about eclipses. Now, where do they get that data? 84 00:09:04,510 --> 00:09:09,470 It turns out almost all of them get it from Southern California. 85 00:09:09,490 --> 00:09:14,770 It's all JPL, the Jet Propulsion Lab. So that is the answer to the question. 86 00:09:15,820 --> 00:09:23,170 All these calculations, mostly more than 90% of them come from the SS. 87 00:09:23,530 --> 00:09:39,980 D is the Solar System Dynamics Group. And this is at the Jet Propulsion Laboratory, JPL, in Pasadena, California. 88 00:09:40,310 --> 00:09:42,410 That's where most of the calculations are done. 89 00:09:42,620 --> 00:09:51,290 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. 90 00:09:51,860 --> 00:09:54,250 The government shutdown has not affected this. 91 00:09:54,260 --> 00:10:00,830 I guess the Jet Propulsion Laboratory is clever enough to insulate its fundings from the whims of the President. 92 00:10:01,940 --> 00:10:10,490 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. 93 00:10:10,700 --> 00:10:14,950 Through here you can get all the details. So just just look at this. 94 00:10:14,960 --> 00:10:22,250 These numbers. Their standard system tracks 791,373. 95 00:10:22,250 --> 00:10:30,110 Asteroids, 3557 comets, 190 planetary satellites, eight planets. 96 00:10:30,120 --> 00:10:36,320 So that answers our question about Pluto, the Sun, various spacecraft and so on. 97 00:10:36,620 --> 00:10:40,459 It's an amazing system using these stern more fairly formulas. 98 00:10:40,460 --> 00:10:44,960 I believe it does this to remarkable accuracy. 99 00:10:44,960 --> 00:10:52,550 They can make predictions thousands of years in the future and thousands of years in the past in order to track historical records. 100 00:10:53,180 --> 00:10:56,270 The accuracies depend on how far away you are in time. 101 00:10:56,630 --> 00:11:02,180 If you're within a few decades, you get unbelievable accuracies in centimetres or metres, 102 00:11:02,180 --> 00:11:06,410 depending on exactly what you're measuring and even a thousands of years away. 103 00:11:06,770 --> 00:11:13,069 You get good accuracy. It's an amazingly mature subject, which for my work, in my opinion, 104 00:11:13,070 --> 00:11:21,260 is really the great example of very high end calculations conducted by the human race, basically. 105 00:11:21,620 --> 00:11:26,249 Very exciting stuff. Now. 106 00:11:26,250 --> 00:11:39,329 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. 107 00:11:39,330 --> 00:11:45,960 They have variable orders, variable step sizes. It's it's a wonderfully mature technology. 108 00:11:46,230 --> 00:11:49,770 Have any of you connected with this stuff to any of your work in such areas? 109 00:11:50,230 --> 00:11:54,050 Uh huh. So what department are you in? Uh huh. 110 00:11:56,000 --> 00:12:03,350 Yeah. Okay. And by the way, I think a lot of the bodies are essentially point masses. 111 00:12:03,350 --> 00:12:08,870 That is good enough for many purposes. The Earth isn't the point mass, but further off planets often are. 112 00:12:09,290 --> 00:12:13,190 Relativistic effects are important. Most of the bodies have that included in them. 113 00:12:14,150 --> 00:12:21,050 So that's a great thing to be aware. So let's say a word about chaos. 114 00:12:25,630 --> 00:12:29,290 And this is a huge subject. 115 00:12:29,440 --> 00:12:34,570 Let me in particular include in my heading the notion of leaping off exponents. 116 00:12:45,390 --> 00:12:50,970 So as I say, chaos has always been there but never really got focussed on until computers came along. 117 00:12:51,180 --> 00:12:54,270 Really in the seventies I guess is when it hit the headlines. 118 00:12:56,170 --> 00:13:00,520 The central question. As people would regard it nowadays. 119 00:13:01,930 --> 00:13:05,200 Four possibly chaotic systems would be this. 120 00:13:05,470 --> 00:13:08,830 If the initial data are perturbed. 121 00:13:12,810 --> 00:13:19,110 And it even makes sense mathematically to talk about infinitesimal perturbations. 122 00:13:20,280 --> 00:13:23,430 Of course, in practice, no perturbation will be infinitesimal. 123 00:13:25,160 --> 00:13:28,340 But mathematically, you can imagine an infinitesimal perturbation. 124 00:13:28,340 --> 00:13:34,130 And you can ask how fast? Does it grow with time? 125 00:13:40,240 --> 00:13:46,150 And the advantage of infinitesimal perturbations is that you can let time go as long as you want, and they're still infinitesimal. 126 00:13:47,740 --> 00:13:55,780 The big distinction is between systems where perturbations grow algebraically and perturbations where systems where they grow exponentially. 127 00:13:56,020 --> 00:14:03,760 So algebraic growth means of time to some power. 128 00:14:05,340 --> 00:14:19,850 Exponential growth. Means O of E to some number like lambda T, where lambda is bigger than zero, 129 00:14:20,360 --> 00:14:24,230 or the real part of lambda is bigger than zero, depending exactly what you're measuring. 130 00:14:25,660 --> 00:14:30,400 Now a chaotic system needs to have this second property. That's not all it needs. 131 00:14:30,610 --> 00:14:34,950 For example. Here's an ode to EU Prime equals U. 132 00:14:35,790 --> 00:14:40,650 The solution is U of T equals either the t. 133 00:14:42,330 --> 00:14:48,510 That solution grows exponentially. And moreover, if you perturb the initial condition, that perturbation will grow exponentially. 134 00:14:48,510 --> 00:14:57,600 But of course, that's not interesting. The interesting cases are where you get exponential growth with a global bounded ness of some kind. 135 00:14:57,780 --> 00:15:02,550 So this is the situation where chaos turns up. 136 00:15:03,180 --> 00:15:13,500 You have exponential growth of perturbations, but still you have in some sense some kind of approximation to global bounded ness of orbits. 137 00:15:14,710 --> 00:15:17,740 Now, I'm not trying to give a precise definition. 138 00:15:17,740 --> 00:15:23,510 And in fact. Although chaos does have precise definitions, 139 00:15:23,970 --> 00:15:30,060 this is not an area where people have settled very happily on one particular definition that everybody uses. 140 00:15:30,300 --> 00:15:34,080 It's a bit complicated, but you know it when you see it, as they say. 141 00:15:36,790 --> 00:15:39,890 Probably most of you know about STROGATZ, his book. 142 00:15:39,910 --> 00:15:41,630 So here's one of the editions of it. 143 00:15:41,650 --> 00:15:50,530 Steve STROGATZ from Cornell was an early person writing a text book, and it's just a fantastic book that everybody loves. 144 00:15:51,280 --> 00:15:57,550 So, Steve STROGATZ, if you want to know about chaos, that's the place to start. 145 00:15:58,810 --> 00:16:02,020 I'm curious. Raise your hand if you were aware of STROGATZ this book. 146 00:16:02,560 --> 00:16:07,030 Yeah, that's impressive. Raise your hand if you've looked at. Have his videos online about chaos. 147 00:16:07,510 --> 00:16:13,460 Okay. He's also a person in the news. 148 00:16:13,490 --> 00:16:17,870 He writes columns for the New York Times and for other magazines and so on. 149 00:16:17,870 --> 00:16:22,010 So he's one of the well known applied mathematicians on the planet. 150 00:16:24,090 --> 00:16:33,900 Just to say a word about the history. So I mentioned John Kerry, who was very important in the pre-history, if you like. 151 00:16:34,110 --> 00:16:37,260 And that's going back to the nine, you know, 1900 ish. 152 00:16:38,300 --> 00:16:42,290 He was concerned with the Three-body problem and the solar system and so on, 153 00:16:42,290 --> 00:16:48,200 and he realised that effectively chaotic things could happen, though he didn't coin that term. 154 00:16:49,010 --> 00:16:53,150 Some other key names were in the 1970s. 155 00:16:54,140 --> 00:17:04,970 There was Roussel and Tompkins and there was Feigenbaum was a key person, so-called universality. 156 00:17:05,600 --> 00:17:14,390 And then there was Bob May, who became Lord May, who showed chaos in biological systems. 157 00:17:14,900 --> 00:17:20,270 And there was Jim York from Maryland who invented the term. 158 00:17:25,180 --> 00:17:29,770 But then the really famous original person was Ed Lorenz. 159 00:17:30,280 --> 00:17:33,420 So. Ed Lawrence at MIT. 160 00:17:35,430 --> 00:17:44,070 And was it 1963 wrote the great paper that sort of put chaos on the map, even though the name wasn't there yet. 161 00:17:44,400 --> 00:17:47,670 He died about ten years ago. He lived to be 90 years old. 162 00:17:48,180 --> 00:17:53,550 I checked this morning on the Web, you get 453 million hits on the word chaos. 163 00:17:53,940 --> 00:18:08,730 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. 164 00:18:08,850 --> 00:18:11,430 This is really a great paper. It's so readable. 165 00:18:11,640 --> 00:18:17,720 If you look at it, you just feel inspired for how much he saw using the primitive computers of those days. 166 00:18:17,730 --> 00:18:19,050 It's it's really great. 167 00:18:19,230 --> 00:18:27,690 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. 168 00:18:29,640 --> 00:18:35,130 So Ed Lorenz. Let me give some examples. 169 00:18:49,150 --> 00:18:54,490 As I say, it's pretty hard not to find chaos if you look at a big dynamical system. 170 00:18:54,700 --> 00:18:59,770 So one field in which people look for chaos is what they call billiards. 171 00:19:00,340 --> 00:19:07,149 Billiards means bouncing particles. In the simplest case, you have a billiard table and you say, 172 00:19:07,150 --> 00:19:14,470 What about the orbits of a ball bouncing around that table if the billiards are on a rectangular table? 173 00:19:18,350 --> 00:19:23,900 That's not chaotic. The perturbations grow only algebraically. 174 00:19:26,180 --> 00:19:36,680 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. 175 00:19:42,420 --> 00:19:53,549 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, 176 00:19:53,550 --> 00:19:58,310 which relates to eigen functions of the in the plus operator in quantum mechanics. 177 00:19:58,320 --> 00:20:02,370 It's a highly developed field. So billiards is one example. 178 00:20:03,810 --> 00:20:07,050 Here's another example if you have any linear equation. 179 00:20:09,130 --> 00:20:12,400 No matter how many variables, it's not chaotic. 180 00:20:16,940 --> 00:20:23,630 It can very easily have exponential growth of perturbations, but that would be in the context of actual solutions going off to infinity. 181 00:20:23,810 --> 00:20:27,950 You don't have the bounded mass needed for things to be really complicated. 182 00:20:29,930 --> 00:20:38,610 Another thing that is never chaotic. Is any one variable or two variable system. 183 00:20:40,820 --> 00:20:44,150 First order autonomous ODI. 184 00:20:48,760 --> 00:20:52,120 So such a problem cannot be chaotic. 185 00:20:52,120 --> 00:20:56,470 An example would be the Vanderpoel equation that we've looked at. And the reason is. 186 00:20:59,770 --> 00:21:04,870 If you have, let's say, two variables, your trajectories are trapped in a plane. 187 00:21:05,110 --> 00:21:08,560 They're completely described by a phase plane. So. 188 00:21:09,630 --> 00:21:13,940 Suppose you're in a plane. There are theorems about this. 189 00:21:13,950 --> 00:21:19,709 Of course. It's impossible to get tangled up in a plane because you can't cross a trajectory. 190 00:21:19,710 --> 00:21:23,970 An autonomous system is completely determined by its value at a point. 191 00:21:24,240 --> 00:21:28,460 So you could have trajectories that look like this, but notice how constrained they are. 192 00:21:28,470 --> 00:21:32,910 They can't really. Since they can't cross, they can't really get tangled. 193 00:21:33,300 --> 00:21:39,600 So there's the hand-waving argument that two dimensional systems cannot be chaotic. 194 00:21:39,930 --> 00:21:43,560 But as soon as you have three dimensions, then chaos is very common. 195 00:21:45,030 --> 00:21:50,780 Now. What about the weather on planet Earth? Which was. 196 00:21:53,240 --> 00:21:57,120 Lawrence's motivation. Well, surely that is chaotic. 197 00:21:57,150 --> 00:22:01,290 It's a bit hard to say because, of course, that's not a well-defined problem. 198 00:22:01,290 --> 00:22:08,820 It's an infinitely complicated system. So, properly speaking, chaos applies or doesn't apply to a precise set of equations, 199 00:22:09,060 --> 00:22:14,400 but many reasonable sets of equations that model Earth are indeed chaotic. 200 00:22:17,830 --> 00:22:21,370 A famous one is turbulent flow in a pipe. 201 00:22:26,980 --> 00:22:30,250 Now it's obvious that, loosely speaking, that's chaotic. 202 00:22:30,250 --> 00:22:32,680 It's very hard to predict crazy things happen. 203 00:22:33,190 --> 00:22:42,610 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. 204 00:22:42,910 --> 00:22:50,740 And there are those who have argued that. What looks like chaos in a turbulent flow. 205 00:22:51,840 --> 00:23:00,540 Is actually, in theory, always transient. So it will look chaotic for a time like our planets that eventually ionise, 206 00:23:00,540 --> 00:23:08,470 but then eventually it has to become a smooth flow again eventually can mean on a doubly exponential timescale. 207 00:23:08,490 --> 00:23:12,060 So it's this is spectacularly academic, that discussion. 208 00:23:12,390 --> 00:23:20,670 Obviously in practice it's chaotic, but it's not completely clear that in an infinite circular pipe modelled by the navier-stokes equations, 209 00:23:20,670 --> 00:23:27,350 technically speaking, one has chaos. Another example. 210 00:23:28,960 --> 00:23:32,920 Now. I guess that's all I have on my list. But let's say a little bit about planets. 211 00:23:36,140 --> 00:23:43,200 About the solar system. So the solar system has a lot of particles in it. 212 00:23:44,710 --> 00:23:49,100 So you would think, given that even three particles, it would seem, can be chaotic. 213 00:23:49,120 --> 00:23:55,540 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. 214 00:23:56,200 --> 00:24:01,090 So there are a lot of epsilon in the solar system compared with the big sun in the middle. 215 00:24:01,720 --> 00:24:04,810 So there's actually some controversy here, too. 216 00:24:06,120 --> 00:24:09,660 But I think most people now believe it is chaotic. 217 00:24:11,810 --> 00:24:17,510 So it would seem that with timescales on the order of tens of millions of years or more. 218 00:24:26,590 --> 00:24:28,270 Now, what does it mean to be chaotic? 219 00:24:28,840 --> 00:24:35,350 Basically, the sort of question that arises is might the solar system eject one of the planets out to infinity in principle? 220 00:24:35,530 --> 00:24:42,460 And I think many people now think that perhaps yes, the answer in principle is yes, though it won't happen in a hurry. 221 00:24:43,270 --> 00:24:48,970 Let me mention some key names here. Just two pairs of names in the 1990s. 222 00:24:51,450 --> 00:24:59,930 Sussman and Widom at MIT. Didn't major calculations which sort of put this problem back on the map after Poincare. 223 00:25:01,040 --> 00:25:05,360 And then more recently in France, the big names, a big name really is Lashkar. 224 00:25:05,370 --> 00:25:20,480 And another key person is Gastineau. Okay. 225 00:25:20,490 --> 00:25:24,960 I guess that's all I'll say. Well, I'll say one more thing. 226 00:25:26,680 --> 00:25:31,820 Chaos is a phenomenon, but also it's something of interest in engineering and people. 227 00:25:32,020 --> 00:25:34,480 It's such a huge thing. People have done everything with it. 228 00:25:34,720 --> 00:25:45,070 For example, if a system is nearly random, then it should be that if you want to achieve a particular outcome, 229 00:25:45,820 --> 00:25:49,240 some set of initial conditions in principle will do that. 230 00:25:49,420 --> 00:25:55,750 Like I said, might there be initial conditions that at times 200 made those planets collide? 231 00:25:56,110 --> 00:26:00,280 And so this would be the general subject that people might call control of chaos, 232 00:26:00,280 --> 00:26:04,810 where you're using the exponential divergence to achieve interesting things. 233 00:26:04,990 --> 00:26:15,670 It's been used in cryptography, for example. You can interplay between the deterministic and the seemingly random aspects to encode messages. 234 00:26:16,420 --> 00:26:24,110 So it was a huge subject. Now I want to say something a little more precise. 235 00:26:24,110 --> 00:26:33,140 So let's talk about one of the favourite examples, which is the Lorenz equations, and I'll call that the next example. 236 00:26:34,340 --> 00:26:37,700 These equations are often the first place that people see chaos. 237 00:26:38,030 --> 00:26:45,440 But for my money, planets orbiting around each other are really the most obvious example, as in what we just looked at. 238 00:26:47,080 --> 00:26:50,170 But let's call this section 4.10. 239 00:26:56,310 --> 00:27:00,420 So these equations go back to 1963. 240 00:27:02,070 --> 00:27:06,720 Now you know that to get chaos, you need at least three variables and you need to be non-linear. 241 00:27:06,870 --> 00:27:09,960 And sure enough, there are three variables and it's non-linear. 242 00:27:10,200 --> 00:27:14,070 So in some sense this is as simple as it can get and still be chaotic. 243 00:27:14,700 --> 00:27:18,690 And let me write down the most famous version of these equations. 244 00:27:18,990 --> 00:27:22,860 X prime equals ten Y minus ten x. 245 00:27:23,790 --> 00:27:27,330 That's a linear equation for how X evolves. 246 00:27:28,260 --> 00:27:32,940 Why Prime equals 28 x. 247 00:27:33,950 --> 00:27:37,610 Minus Y, minus x, z. 248 00:27:38,030 --> 00:27:45,440 So that's a nonlinear equation for how Y evolves. And Z prime is x. 249 00:27:46,720 --> 00:27:51,400 Times Y minus eight thirds. Time Z. 250 00:27:52,660 --> 00:27:54,670 So that's another non-linear equation. 251 00:27:55,090 --> 00:28:01,240 Notice the nonlinear nonlinear sureties are as simple as you could ask for just quadratic products of two terms. 252 00:28:02,800 --> 00:28:08,740 The precise numbers can be changed. It doesn't have to be exactly 28 and 8/3 and ten. 253 00:28:09,040 --> 00:28:17,620 But these particular choices have become so standard that 99% of discussions of the Lorenz equations use these numbers. 254 00:28:18,070 --> 00:28:23,410 So if you start analysing this, what you find is that one solution? 255 00:28:26,140 --> 00:28:30,160 Is X equals Y equals z equals zero. 256 00:28:31,840 --> 00:28:36,460 Plug that in. And you see, of course, the right hand sides are all zero. 257 00:28:36,910 --> 00:28:40,000 Therefore, the equation is satisfied. 258 00:28:40,180 --> 00:28:44,830 So this would be called a fixed point of this dynamical system. 259 00:28:46,930 --> 00:28:51,400 A fixed point is a point which doesn't vary with time steady state. 260 00:28:52,090 --> 00:28:58,390 It's an unstable fixed point, but it is a fixed point. There are two more fixed points. 261 00:29:01,520 --> 00:29:09,800 And they are at x equals y equals plus or minus six square root of two. 262 00:29:11,580 --> 00:29:22,780 And Z equals 27. So if you analyse those, plug these into the equation, you'll find that again, the derivatives are all zeros. 263 00:29:22,780 --> 00:29:30,310 So it's a fixed point. It's unstable, meaning that small perturbations will grow away from those fixed points. 264 00:29:30,550 --> 00:29:35,770 So when something's unstable, in a physical sense, you would expect not to observe it in the real world. 265 00:29:38,030 --> 00:29:46,730 Now what happens is that when you compute most solutions to these equations, they move around in a seemingly random fashion. 266 00:29:50,520 --> 00:29:55,380 Lorenz claimed that in 1963, many people thought he was wrong. 267 00:29:56,580 --> 00:30:01,560 They argued that it was errors in the computer. You know, this is 1963 computers. 268 00:30:01,590 --> 00:30:10,010 Just imagine how slow and moreover, the floating point arithmetic in those days was not as established as now. 269 00:30:10,020 --> 00:30:13,380 Every machine was different. People didn't understand it so well. 270 00:30:14,250 --> 00:30:19,380 It was hard for him to persuade people that it was true. But he was very careful and he did persuade people. 271 00:30:20,190 --> 00:30:26,330 It took 36 years. For a proof. 272 00:30:27,200 --> 00:30:35,180 So there's a famous guy called Warwick Tucker, who I guess is the head of the math department at the University of Uppsala. 273 00:30:36,650 --> 00:30:41,540 And he wrote a paper called The Lorenz Attractor Exists. 274 00:30:43,870 --> 00:30:47,020 So this is the gulf between pure and applied mathematics. 275 00:30:49,950 --> 00:30:58,830 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. 276 00:30:59,520 --> 00:31:04,410 He proved that there is a strange attractor, as they call it, for the Lorenz equation. 277 00:31:04,420 --> 00:31:12,420 And this was in a famous French journal in 1999. 278 00:31:16,000 --> 00:31:20,580 What's the. Layup on off exponent. 279 00:31:23,120 --> 00:31:27,950 I didn't define it, did I. But the layup on off exponent which is in my heading. 280 00:31:30,670 --> 00:31:40,930 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, 281 00:31:41,890 --> 00:31:48,040 so it's the maximum exponential rate at which perturbations can grow infinitesimal perturbations. 282 00:31:48,280 --> 00:31:57,400 These things are hard to calculate. I think for the Lorenz system it's thought to be about 0.9057. 283 00:31:59,770 --> 00:32:05,020 So about one. So perturbations grow on average at a rate of about E to the T. 284 00:32:06,610 --> 00:32:18,260 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. 285 00:32:18,280 --> 00:32:21,340 It's a set in three dimensions. 286 00:32:21,730 --> 00:32:25,030 The reason they're three dimensions is that we have X and Y and Z. 287 00:32:28,640 --> 00:32:32,840 And so the strange attractor is a three dimensional object. 288 00:32:35,170 --> 00:32:36,910 But one of the wonderful things. 289 00:32:38,650 --> 00:32:48,670 About chaos is that these are geometric objects, although they sit in three dimensions, may have a lower dimensionality in a precise sense. 290 00:32:48,880 --> 00:32:54,640 And I believe that the dimension of the strange attractor is known to quite a few digits. 291 00:32:55,330 --> 00:32:58,420 It's something like 2.0. 292 00:32:59,550 --> 00:33:03,840 6 to 7 1600. 293 00:33:04,860 --> 00:33:11,580 I'm going to plot it in a second. So what that means is that the chaotic orbits of the Lorenz equations. 294 00:33:13,700 --> 00:33:21,679 After an initial period, essentially lie on a object which is nearly two dimensional, but slightly more so. 295 00:33:21,680 --> 00:33:25,340 It's a little thicker than a two dimensional manifold. 296 00:33:26,090 --> 00:33:42,889 So let's play with that now. Okay. 297 00:33:42,890 --> 00:33:46,070 There are lots of ways to play with the Lorenz equations. Everybody's got a method. 298 00:33:46,070 --> 00:33:52,460 So let's start with Matt Labs method. If I type Lorenz in MATLAB, I there's a built in demo. 299 00:33:53,090 --> 00:33:56,890 So let's see what that does. This has been there for ages. 300 00:33:56,900 --> 00:34:00,260 I'll press start and you see it just. 301 00:34:02,960 --> 00:34:07,460 Uses Odie something or other. I don't know which one to compute a trajectory. 302 00:34:08,120 --> 00:34:14,810 And you can see the this is the strange attractor emerging as you have this chaotic motion. 303 00:34:15,260 --> 00:34:21,520 You see it sort of looks two dimensional. It's got these two butterfly wings, if you like, but they're not quite flat. 304 00:34:21,530 --> 00:34:22,819 If you rotated around, 305 00:34:22,820 --> 00:34:31,220 you'd see there's a little bit of thickness there and there is a precise way that mathematicians measure these non integer dimensions. 306 00:34:32,270 --> 00:34:39,050 Any comments or questions. Stop. 307 00:34:42,930 --> 00:34:46,620 Let's do another demo. Let's try and check. 308 00:34:46,670 --> 00:34:54,640 Fine. I'll use a cheap, gooey. So the graphical user interface to the O.D initial value code. 309 00:34:54,660 --> 00:34:59,110 So there we are, one of the built in demos is. 310 00:35:00,350 --> 00:35:06,060 The Lorenz equations. This is a much slower way to solve the problem. 311 00:35:06,420 --> 00:35:11,310 Telephone is never the fastest, so it's a coupled initial value problem, a system. 312 00:35:11,790 --> 00:35:23,880 And I'll click on Lorenz Equations. And then if I type solve, you can see it's only going up to time 15. 313 00:35:24,180 --> 00:35:30,810 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. 314 00:35:31,530 --> 00:35:35,190 And you can see this familiar random looking behaviour. 315 00:35:38,510 --> 00:35:44,600 One of the nice things about a system like this is that you can play around and change things very easily. 316 00:35:45,380 --> 00:35:50,190 So what happens if you change some of those coefficient ten or 28 or 8/3? 317 00:35:50,480 --> 00:35:53,810 I'll just make one change that I happen to know does something interesting. 318 00:35:54,170 --> 00:36:09,650 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. 319 00:36:09,890 --> 00:36:22,850 So that's not a chaotic system, evidently. Let me also draw your attention to the book Exploring Odes, which I mentioned. 320 00:36:26,070 --> 00:36:30,420 So if you look in there, of course, there's a chapter on Chaos, chapter 13, I think. 321 00:36:30,630 --> 00:36:36,990 But then if you go to this appendix, I like so much appendix B with the more examples. 322 00:36:38,960 --> 00:36:44,000 A bunch of those examples are chaotic systems, so I think it starts with 64. 323 00:36:46,710 --> 00:36:53,870 So in fact, when we were writing the book. I asked myself whether. 324 00:36:58,350 --> 00:37:07,200 Please. Can you tell if this is working? Yeah. I asked myself whether you might have a scalar, chaotic system. 325 00:37:07,410 --> 00:37:14,090 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. 326 00:37:14,760 --> 00:37:18,870 So sure enough, if you look on the web for scalar chaotic systems, you find one. 327 00:37:19,020 --> 00:37:20,850 And this is a simplified version of that. 328 00:37:21,000 --> 00:37:28,280 Notice is a third derivative here and we've cooked it up so that you're getting chaotic behaviour in this scalar ODI. 329 00:37:29,610 --> 00:37:32,759 If you look at the other examples, I won't linger. 330 00:37:32,760 --> 00:37:37,320 But so there you have a forced nonlinear pendulum which appears in some books. 331 00:37:38,340 --> 00:37:45,870 Here you have the Roessler equations. This particular orbit is periodic, but you can see a kind of a period doubling phenomenon. 332 00:37:46,140 --> 00:37:53,430 With some parameter it would have been simply periodic. Now you can see the two different amplitudes reflect a doubling of the period. 333 00:37:53,820 --> 00:37:59,040 This is the universal so-called root chaos that Feigenbaum made famous. 334 00:38:00,430 --> 00:38:03,490 Here's a three body problem, just as we've been looking at. 335 00:38:04,210 --> 00:38:08,980 This one is not interesting. The initial conditions don't do anything remarkable. 336 00:38:09,970 --> 00:38:12,970 This one, on the other hand, is interesting, chaotic. 337 00:38:14,980 --> 00:38:21,130 Here's a double pendulum. That's a nice example that you often see in people's offices because you can build one physically. 338 00:38:22,920 --> 00:38:25,920 Here's the so-called hand on health equations. 339 00:38:27,200 --> 00:38:35,390 And I guess that's it for the chaos. Now. 340 00:38:38,710 --> 00:38:42,460 I thought I had an example that showed the growth of perturbations. 341 00:38:43,030 --> 00:38:47,049 Was that the very first one? Did I fail to? Where am I? 342 00:38:47,050 --> 00:38:58,170 Here. Let me remind myself if I do em again. 343 00:39:00,430 --> 00:39:04,690 No, that's not the one I want. Uh, I don't think it is. 344 00:39:04,840 --> 00:39:09,180 Well. I'll run that while I figure out what I'm looking for. 345 00:39:27,610 --> 00:39:32,320 No, I guess it's just in the next one I'm going to show you. I thought I had one already. 346 00:39:36,100 --> 00:39:44,020 Okay. So let me finish up by discussing another famous example of chaos in the billiard department. 347 00:40:12,680 --> 00:40:16,549 So if you think of billiards, the physics is so simple. 348 00:40:16,550 --> 00:40:21,630 The particle is always going at a fixed speed. So how many variables are there? 349 00:40:21,650 --> 00:40:25,640 Well, at any time it has a position, and that's two variables. 350 00:40:26,180 --> 00:40:34,580 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. 351 00:40:40,030 --> 00:40:45,400 And the example I want to mention is called the senior billiard. 352 00:40:51,600 --> 00:41:03,549 So this is our last example. Seni is a famous mathematician and physicist billiard. 353 00:41:03,550 --> 00:41:08,200 And I'm also going to mention the so-called Siam 100 degree challenge. 354 00:41:16,550 --> 00:41:21,290 Which I think came up earlier when we talked about optimisation in two dimensions. 355 00:41:22,820 --> 00:41:25,250 So if you look at the handout, you'll see that the. 356 00:41:26,840 --> 00:41:38,809 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, 357 00:41:38,810 --> 00:41:42,470 each to ten digits of accuracy. So these are the ten problems. 358 00:41:43,790 --> 00:41:53,030 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. 359 00:41:53,570 --> 00:41:57,140 However, one of the problems is chaotic, and that's number two. 360 00:41:58,040 --> 00:42:01,100 So if you look at problem number two. 361 00:42:03,590 --> 00:42:10,760 You see, the way it's described there is that you have an infinite set of circles in the plane. 362 00:42:14,110 --> 00:42:18,130 And you have a particle moving around and a particular. 363 00:42:19,080 --> 00:42:25,780 Orbit is specified doing things like this. Well, that's a chaotic system. 364 00:42:26,350 --> 00:42:30,670 It's a billiard problem in which the boundary is not just straight lines, but curves. 365 00:42:31,180 --> 00:42:38,380 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. 366 00:42:38,800 --> 00:42:43,660 What Seni did was realise that the simplest version of this would be. 367 00:42:45,380 --> 00:42:48,800 That kind of domain. Because if a particle. 368 00:42:51,210 --> 00:42:55,620 Keeps going. Well, that's equivalent essentially to being reflected. 369 00:42:55,860 --> 00:43:06,030 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. 370 00:43:06,480 --> 00:43:13,770 So that is his billiard. And he proved that the motion is chaotic. 371 00:43:15,120 --> 00:43:24,989 And I want to show you that in action. So this is a code that I actually haven't printed up, though. 372 00:43:24,990 --> 00:43:28,770 I'll put it on the web called M 30 Circles. 373 00:43:36,670 --> 00:43:42,660 So suppose I say am 30 circles. And. 374 00:43:45,090 --> 00:43:51,840 It asks how much of a perturbation I want. So I'll start with a perturbation of ten to the minus. 375 00:43:52,770 --> 00:43:55,890 Ten and you'll see as we run it what that means. 376 00:43:56,490 --> 00:43:57,870 I'll tell you before we run it, 377 00:43:58,470 --> 00:44:08,520 we're going to track the 100 digit challenge problem with the specified initial condition and then also with a perturbed initial condition. 378 00:44:08,730 --> 00:44:12,020 And we're going to compare the two orbits. So. 379 00:44:13,860 --> 00:44:17,250 You can see that's actually two particles bouncing around. 380 00:44:17,970 --> 00:44:20,970 And then finally, after a while, they become distinguishable. 381 00:44:21,150 --> 00:44:25,680 The red one has gone off. Who knows where? And the black one is still there. 382 00:44:26,160 --> 00:44:31,290 So. Let's press. No, I'm going to run it once more before pressing. 383 00:44:31,290 --> 00:44:36,310 Enter. Okay. Let's try a different perturbation. 384 00:44:36,320 --> 00:44:40,010 I'll try ten to the -12 now. 385 00:44:40,040 --> 00:44:49,370 10 to -8. We keep losing the particle. 386 00:44:49,380 --> 00:44:53,400 I want to find one that where we keep the red particle. So I try ten to the minus nine. 387 00:44:55,760 --> 00:44:59,180 Must be possible a. It keeps. 388 00:45:00,660 --> 00:45:12,910 Okay. That's a little better. 389 00:45:12,940 --> 00:45:18,430 Okay, good. I like that one more. So you can see that for a certain time. 390 00:45:18,670 --> 00:45:25,240 Although the trajectories are diverging exponentially, they're so close that you don't see that in the in the picture. 391 00:45:25,420 --> 00:45:28,420 But then eventually they have diverged so much that. 392 00:45:29,420 --> 00:45:35,570 They're no longer correlated in any way. Now what happens at that point is that. 393 00:45:36,670 --> 00:45:40,040 The exponential divergence ceases. Everything is now algebraic. 394 00:45:40,110 --> 00:45:44,680 Nothing is growing exponentially in this picture. Apart from those initial perturbations. 395 00:45:44,890 --> 00:45:47,800 And that's when I press return the second plot that comes up. 396 00:45:48,070 --> 00:45:53,530 So this plot shows the distance between the red and the black dots as a function of time. 397 00:45:53,740 --> 00:46:00,370 And we started with perturbation ten to the minus nine. And you see a pretty systematic exponential growth. 398 00:46:00,610 --> 00:46:09,220 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 00:46:10,030 --> 00:46:16,150 So you see, mathematically, if we started with an infinitesimal perturbation, this behaviour in principle would go forever. 400 00:46:18,510 --> 00:46:21,990 Let's do it one more time with a very small perturbation. Suppose I say. 401 00:46:24,300 --> 00:46:29,250 And 30. I'll say one E -15 and see what happens. 402 00:46:39,060 --> 00:46:46,500 Okay. So there you can see the exponential growth is just barely beginning to level off at the end. 403 00:46:46,500 --> 00:46:54,600 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 00:47:01,200 --> 00:47:10,850 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 00:47:11,480 --> 00:47:15,290 I think the essential answer is, no, there isn't. You just play with it. 407 00:47:15,290 --> 00:47:20,090 And of course, play is a loaded word, but you do very serious computing. 408 00:47:20,780 --> 00:47:25,240 But no. And this is why, for example, the solar system is controversial. 409 00:47:25,250 --> 00:47:30,590 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.