1 00:00:14,430 --> 00:00:19,560 Thank you for that kind introduction and it's great to be here. You all for coming this evening? 2 00:00:20,220 --> 00:00:25,740 I'm going to start by, as Eileen says, winding the clock back a bit and in fact, 3 00:00:26,220 --> 00:00:34,470 start by referring to what was an extremely what has proved to be an extremely influential popular science book, 4 00:00:34,470 --> 00:00:43,200 which was published in the second half of the 1980s by James Gleick, because it was this book that coined this iconic phrase, 5 00:00:43,200 --> 00:00:50,670 The Butterfly Effect, which I think has pretty much gone into everyday language and popular culture. 6 00:00:52,770 --> 00:00:57,420 In fact, the very first chapter of the book is called The Butterfly Effect. 7 00:00:57,900 --> 00:01:06,690 And in this chapter, Glick describes the work of MIT meteorologist Ed Lorenz, whose picture is up there, 8 00:01:07,200 --> 00:01:15,990 and a paper that he wrote back in 1963, which was pretty much ignored, I would say, for ten years or so. 9 00:01:16,380 --> 00:01:22,410 It was published in what most scientists think of as an obscure metrological journal. 10 00:01:23,040 --> 00:01:26,050 Turns out, actually it's the premier meteorological journal in the field. 11 00:01:26,100 --> 00:01:34,140 Anyway, there we go. But what Gleick describes, and I'm going to talk about this at some length in the talk, 12 00:01:34,950 --> 00:01:46,499 is how Lorenz came to discover three very simple equations which had no determinism or no randomness. 13 00:01:46,500 --> 00:01:52,200 Everything was precisely defined. The three equations coupled together in a way which I'll tell you. 14 00:01:52,920 --> 00:02:03,149 And it had the property that it was almost impossible to predict the future of these equations, in the sense that starting with just a tiny, 15 00:02:03,150 --> 00:02:09,960 tiny, tiny perturbation, let's say, to the initial state, the solution developed quite differently. 16 00:02:10,470 --> 00:02:14,370 So let me actually illustrate that right at the beginning with this animation. 17 00:02:14,370 --> 00:02:17,669 So we're going to look at the evolution in time of what actually are. 18 00:02:17,670 --> 00:02:25,290 Two, it's not apparent until about now that there are actually two different solutions of these equations. 19 00:02:25,740 --> 00:02:31,530 Starting as I say, from almost but not quite identical initial conditions. 20 00:02:32,040 --> 00:02:38,950 And after a while, although they track together the red and the blue for a while, they sort of correlate. 21 00:02:39,690 --> 00:02:48,060 And so knowing one of the solutions, if you like, tells you almost nothing about the second solution after a certain time. 22 00:02:50,190 --> 00:03:01,260 Now, Glik then went on to say that this model of Lorenz basically formed a, you know, 23 00:03:01,440 --> 00:03:09,059 a hard quantitative bedrock for his notion that he'd been that he Lawrence had 24 00:03:09,060 --> 00:03:16,300 been talking about for some years of how the flap of a butterfly wings could go, 25 00:03:16,350 --> 00:03:21,090 let's say, in Brazil, could cause a tornado, let's say, in Texas. 26 00:03:22,170 --> 00:03:27,749 Now, this is where the history actually becomes slightly confused. 27 00:03:27,750 --> 00:03:30,990 And one of the. So I've got two parts to my talk. 28 00:03:31,440 --> 00:03:34,860 One is to try to review what is pretty standard stuff, 29 00:03:34,860 --> 00:03:45,450 which is how Lorenz came to arrive at these equations and some of the amazing insights he had, incidentally, in linking these very simple, 30 00:03:45,960 --> 00:03:54,480 essentially equations that Isaac Newton would have understood very well with modern 20th century fractal geometry, 31 00:03:54,480 --> 00:04:04,620 a sort of concept that Newton would have found quite alien. So there's enormous insights that Lorenz had in deriving his 1963 model. 32 00:04:06,060 --> 00:04:14,400 But I also want to say, and this is something which I'm sure most most of you will not know is that actually Glick got it completely 33 00:04:14,400 --> 00:04:22,950 wrong about attributing this 63 model to the idea that Lorenz had in when you referred to the butterfly effect. 34 00:04:23,310 --> 00:04:26,790 So unfortunately, the butterfly effect has actually been misnamed. 35 00:04:26,880 --> 00:04:29,250 And I'm going to try to explain why that is. 36 00:04:31,240 --> 00:04:39,910 So the term the term well, the term the butterfly effect comes from Glick's book, but essentially refers to this paper or this talk, 37 00:04:39,910 --> 00:04:48,370 let's say, that Lorenz gave in 1972 at the American Association for the Advancement of Science and the title. 38 00:04:48,490 --> 00:04:55,150 It was a kind of a sort of semi it was essentially a public talk about predictability of weather. 39 00:04:56,110 --> 00:05:00,520 And that was the title. Does the flap of a butterfly wings in Brazil set off a tornado in Texas? 40 00:05:01,150 --> 00:05:06,250 So what I wanted to do in the sort of towards the end of the talk is actually 41 00:05:06,700 --> 00:05:13,480 to to say a bit more about precisely what Lorenz said in this 1972 meeting. 42 00:05:13,810 --> 00:05:21,910 And you will see that it's actually not referring to his 63 paper at all, but to something he published, in fact, in 1969. 43 00:05:21,910 --> 00:05:26,530 And that is much less well known, I would say, in the broad community. 44 00:05:27,160 --> 00:05:35,380 And in fact, this problem that Lorenz Lorenz wrote about in the 69 paper and which he talked about in the 72 meeting, 45 00:05:35,710 --> 00:05:43,360 turns out to be one of the great unsolved or closely linked one of the great unsolved problems in 21st century mathematics. 46 00:05:44,050 --> 00:05:51,370 So it actually gets to the heart of something which is actually much more radical than kind of chaos theory, 47 00:05:51,370 --> 00:05:55,090 although you might think chaos theory is fairly radical, but this is even yet more radical. 48 00:05:56,620 --> 00:05:58,350 So so that's the outline of the talk. 49 00:05:58,360 --> 00:06:05,410 You'll know some of it maybe, and I'll fill in some details, but towards the end I'll say stuff that I'm sure most of you don't know. 50 00:06:05,590 --> 00:06:08,980 And I hope to I'm not going to really change popular culture. 51 00:06:09,440 --> 00:06:16,890 It'll it will always be called the butterfly effect. But I'm going to call what I what what Lorenz really meant the real butterfly effect. 52 00:06:16,900 --> 00:06:23,799 So we'll come to that towards the end of the talk. Let me just say a little bit about the biography of Lorenz. 53 00:06:23,800 --> 00:06:32,650 So here he is, is as a young man, grew up in New Hampshire, very interested in astronomy as a kid, 54 00:06:33,820 --> 00:06:38,580 apparently had a passing interest in weather, but it wasn't a particularly sort of passion of his. 55 00:06:38,660 --> 00:06:43,240 In fact, probably weather was a nuisance because it obliterated the stars when he wanted to go out and look at them. 56 00:06:44,260 --> 00:06:48,910 But he claims, you know, in in later years, he had he had somewhat of an interest in weather. 57 00:06:49,810 --> 00:06:53,260 But he went to university, he went to study mathematics. 58 00:06:53,950 --> 00:06:59,169 And, in fact, he went to Harvard and went through his undergraduate degree and in fact, 59 00:06:59,170 --> 00:07:04,360 started a postgraduate degree in mathematics under the great George Birkhoff, 60 00:07:04,900 --> 00:07:09,850 where he was studying some topic in apparently remaining in geometry, nothing directly to do with chaos. 61 00:07:10,960 --> 00:07:15,460 And his plan was to carry on. That was going to be his Ph.D. in mathematics. 62 00:07:15,880 --> 00:07:20,590 But then the Second World War came and sort of upset the plan. 63 00:07:21,490 --> 00:07:31,330 And I think by chance, Lorenz discovered an advertisement that the CIA, the Army or the or the Air Force, I think it was the army at the time. 64 00:07:32,230 --> 00:07:38,920 They were looking for weather forecasters. Maybe it's the Air Force because it was two planes and so on. 65 00:07:40,330 --> 00:07:44,920 So he thought, well, okay, well, I've have a thought back to his childhood, a little bit of interest in weather. 66 00:07:44,920 --> 00:07:48,670 So maybe I'll that will be a good sort of career for me during the war. 67 00:07:48,700 --> 00:07:55,509 So he he joined up and, you know, he was sufficiently good that after he'd been through this course, 68 00:07:55,510 --> 00:08:00,129 they decided he was the right person to teach the next lot of weather forecasters to be weather forecasters. 69 00:08:00,130 --> 00:08:08,710 So he actually became a tutor, if you like, in weather forecasting for the for the Air Force. 70 00:08:09,670 --> 00:08:13,749 He eventually got sent out to Okinawa, I think, in the Pacific. 71 00:08:13,750 --> 00:08:19,420 So he actually was involved in some of the real action in the in the Pacific sector. 72 00:08:21,170 --> 00:08:28,549 Came back after the war and thought about, okay, do I restart my career in mathematics and decided, 73 00:08:28,550 --> 00:08:31,550 actually, you take quite an interest in a real interest in Weber. 74 00:08:32,060 --> 00:08:40,550 And so instead he enrolled at MIT just up the road to do a Ph.D. in meteorology. 75 00:08:42,010 --> 00:08:50,140 And the topic he was essentially given was trying to solve this equation. 76 00:08:50,410 --> 00:08:56,890 So it doesn't matter if you don't know this equation, but this is one of the archetypal equations in fluid mechanics. 77 00:08:57,310 --> 00:09:01,690 It's essentially Newton's second law of motion, you know, suit and circular motion force. 78 00:09:01,690 --> 00:09:05,709 It was mass times acceleration. This is well, it's actually the other way around. 79 00:09:05,710 --> 00:09:13,180 It sets knots times acceleration equals force, but written for a fluid, which could be the atmosphere or it could be the oceans, 80 00:09:13,180 --> 00:09:18,190 or it could be a laboratory fluid, a fluid that potentially has many, many, many scales of motion. 81 00:09:19,150 --> 00:09:22,960 And it is actually a remarkable thing that if you count, remember, 82 00:09:23,290 --> 00:09:27,280 the last time I counted the number of symbols, it was around 20 something, 22 or so. 83 00:09:27,520 --> 00:09:34,030 Mathematical symbols is always strikes me as a wonderful thing that with just 22 mathematical symbols, 84 00:09:34,030 --> 00:09:41,559 you can describe the dynamics of every scale of motion in the atmosphere from the very largest jet streams, 85 00:09:41,560 --> 00:09:48,170 you know, which extends thousands of kilometres in length in the upper atmosphere, you know, 86 00:09:48,250 --> 00:09:52,990 right down to clouds, right down to the little eddies coming out of my mouth as I speak. 87 00:09:53,050 --> 00:09:56,200 This is all described by this one set of equations. 88 00:09:56,200 --> 00:10:00,250 So this is Lorenz's job to look at. How do you know the problem is? 89 00:10:01,420 --> 00:10:07,740 So, okay, so this is this is this equation can be likened to very much a work of art. 90 00:10:07,750 --> 00:10:15,260 It is a work of art. But instead of likening it to a Renoir here, I'm going to liken it to a Russian doll. 91 00:10:16,980 --> 00:10:23,390 This is actually a very special Russian doll. Russian dolls unpack into small Russian dolls. 92 00:10:23,780 --> 00:10:30,439 This one unpacks into yet smaller Russian dolls and indeed yet smaller Russian dolls, and indeed yet smaller Russian dolls. 93 00:10:30,440 --> 00:10:33,500 And indeed, yet smaller Russian dolls. Dot, dot, dot. 94 00:10:36,300 --> 00:10:41,760 And in the same way, if you actually want to solve this equation, you have to, as it were, 95 00:10:41,760 --> 00:10:47,340 unpack it also into actually what turned out to be billions of individual equations. 96 00:10:47,760 --> 00:10:50,969 This is actually what makes weather forecasting one of the things that makes it so difficult, 97 00:10:50,970 --> 00:10:56,400 because you need to unpack it into the equations which describe the big jet stream, 98 00:10:56,400 --> 00:11:01,410 which describes low pressure systems, which describe clouds, which describe some cloud turbulence and all that stuff. 99 00:11:03,120 --> 00:11:13,420 So the way to solve these equations, which is what Lorenz did, is, is to look at truncating these equations. 100 00:11:13,440 --> 00:11:24,180 Oh, sorry. Before I do that, yes. Let's let's say that these Russian dolls can be likened to, you know, the worlds or the eddies in a turbulent fluid. 101 00:11:24,480 --> 00:11:33,030 And in fact, it's it's probably appropriate at this stage to refer to the little piece of doggerel that was written by one of the founding. 102 00:11:33,180 --> 00:11:39,900 Well, one of the real pioneers of turbulence theory and also, incidentally, a pioneer of of weather forecasting. 103 00:11:40,290 --> 00:11:44,760 Lewis Fry Richardson, sometime in the beginning of the century. 104 00:11:45,390 --> 00:11:53,160 Anyway, this famous little poem, big whirls of little whirls that feed on their velocity and little wells have lesser wells and so on to viscosity. 105 00:11:54,610 --> 00:12:01,979 Okay. What a great poet. But still, it makes the point that in a turbulent system, you have these many, many, many scales. 106 00:12:01,980 --> 00:12:07,590 So these scales, the big wells or the big Russian doll select little wells, the of the smaller Russian dolls and so on. 107 00:12:08,520 --> 00:12:13,979 And the fact that this notion of feeding on their velocity captures this idea that, as it were, 108 00:12:13,980 --> 00:12:19,350 the Russian dolls can sort of bash into each other and transfer energy from one one Russian doll to another. 109 00:12:19,860 --> 00:12:26,999 And this is manifest mathematically in the fact that the equation here this you is the fluid velocity, 110 00:12:27,000 --> 00:12:36,150 and it sort of multiplies itself and that makes the system nonlinear and not linearity is the thing that allows energy to move up and down the scales. 111 00:12:36,480 --> 00:12:44,809 And that's one of the complications of things. So Lorenz's PhD was actually about trying to whips was actually if I could go back, 112 00:12:44,810 --> 00:12:50,510 was trying to solve these equations in a sort of simplified way by getting 113 00:12:50,510 --> 00:12:55,669 rid of some of the scales and just treating maybe the system in a simplified, 114 00:12:55,670 --> 00:13:04,110 approximate way. And he came up with some what are called time stepping schemes and other types of what are called numerical schemes, 115 00:13:04,110 --> 00:13:10,709 which actually even to this day are still used. Anyway, the study was, I would say, moderately successful. 116 00:13:10,710 --> 00:13:13,830 It wasn't like I set the world on fire, but it was moderately successful. 117 00:13:14,340 --> 00:13:22,050 And he found that within a few years of his PhD, he'd been invited back to MIT as a proper member of faculty. 118 00:13:22,950 --> 00:13:28,170 But there was a slight rider on this appointment because he had to be in charge of a group 119 00:13:28,410 --> 00:13:33,470 that was whose kind of principal research activity was doing long range weather forecasting. 120 00:13:33,480 --> 00:13:37,020 So long range means, you know, a month or two ahead, 121 00:13:37,620 --> 00:13:42,630 whereas the people that were trying to kind of approach weather forecasting from the from 122 00:13:42,690 --> 00:13:45,960 the point of view of solving these navier-stokes equations were only thinking about, 123 00:13:46,430 --> 00:13:50,940 you know, maybe 12 hours ahead at the moment or maybe a day ahead at most, but just a very short time. 124 00:13:52,290 --> 00:13:54,989 So how would people even think about doing long range forecasting? 125 00:13:54,990 --> 00:13:59,040 So this was actually done by a completely different method to solving these equations. 126 00:13:59,040 --> 00:14:07,280 It was just done with the statistics. So the idea is you have a big pile of weather maps which let's say we do this today. 127 00:14:07,290 --> 00:14:13,140 We got a big part of weather maps from today going right back to whenever you like, 1800 for the sake of argument. 128 00:14:15,000 --> 00:14:22,650 So we're in what have we our may. So let's suppose our task is to forecast what the weather's going to be like a month from now. 129 00:14:22,660 --> 00:14:26,040 So the monthly forecast for June or July or something. 130 00:14:27,540 --> 00:14:35,670 So the idea that this group had and incidentally, he got a lot of this group got a lot of support from the Statistics Department at MIT. 131 00:14:35,820 --> 00:14:45,530 So this this this had some pretty high level support was you just go back in time and find a weather map that from the past that looks like today. 132 00:14:45,540 --> 00:14:52,470 So you go back and find, let's say 1961, May 1961, the weather looked very similar to May 2017. 133 00:14:53,490 --> 00:15:04,920 Then what you do, you predict for June, July 2017, what the weather was in May and June of whatever I said, 1961, it's kind of method of analogues. 134 00:15:06,150 --> 00:15:12,990 So Lawrence got put in charge of this team, and his first reaction was, Hmm, this didn't sound right to me. 135 00:15:13,020 --> 00:15:17,639 You know, what's the scientific basis for this? This idea of just so you can find an analogue? 136 00:15:17,640 --> 00:15:20,969 Because it seemed to suggest that the weather was somehow very periodic. 137 00:15:20,970 --> 00:15:30,600 You know, if it if it was safe, it was like it is in May 61, that necessarily what we see in the next couple of months will be what happened in 1961. 138 00:15:30,840 --> 00:15:34,680 And his kind of intuition was that the weather isn't periodic like this. 139 00:15:34,950 --> 00:15:43,110 It seems to be irregular. And apparently had long battles with people not only in this group, but again with the statistics, people that I met. 140 00:15:44,160 --> 00:15:49,620 And he said, you know, I've got to try and sort this out. I'm going to try to prove that this really this idea isn't going to work. 141 00:15:49,800 --> 00:15:56,490 How am I going to do it? So that's when he went back to his truncated navier-stokes equations. 142 00:15:57,030 --> 00:16:02,490 So maybe I can prove that this property of of periodicity just doesn't happen. 143 00:16:02,850 --> 00:16:09,120 If I take a semi realistic model and use I mean, computer technology was, was starting to in the fifties, 144 00:16:09,120 --> 00:16:17,040 computer technology was starting to arise so I could solve these equations on a computer and just show them that this won't work. 145 00:16:18,450 --> 00:16:22,580 So he set about, you know, taking actual weather type of equations. 146 00:16:22,590 --> 00:16:26,280 And for a long time, he worked with many Russian dolls of the eight. 147 00:16:26,310 --> 00:16:31,170 So actually, he worked for a long time with 12, you know, a 12 component. 148 00:16:32,760 --> 00:16:36,899 So truncating the equation down to 12 compares. But, you know, the computers just weren't up to the job. 149 00:16:36,900 --> 00:16:40,950 And he found this was really hard work trying to do anything. 150 00:16:40,950 --> 00:16:43,500 It just took too long to get anywhere. 151 00:16:44,160 --> 00:16:52,739 And he had a real breakthrough when he talked with a colleague of his from a nearby university who had been studying not so much weather, 152 00:16:52,740 --> 00:16:59,940 but what's called convection in a in a fluid where you heat a fluid from below and look at the circulations that develop. 153 00:17:00,900 --> 00:17:09,960 And this guy actually said that he had a seven component model that seemed to seem to have this property of non periodicity. 154 00:17:10,480 --> 00:17:16,410 Maybe Lorenz could look at that and Lorenz did in fact move on to that problem and 155 00:17:16,410 --> 00:17:21,720 he realised within this subset of seven equations there was actually a three. 156 00:17:22,110 --> 00:17:29,190 So he realised when he was looking at these non periodic solutions that four of these components almost went to zero. 157 00:17:29,200 --> 00:17:32,330 So we kind of had the intuition that there would be a three component. 158 00:17:32,340 --> 00:17:36,330 He had a three component subset that. 159 00:17:39,760 --> 00:17:47,790 That had the required property. And this is what led him to these iconic equations, to Lorenz 63 equations. 160 00:17:47,790 --> 00:17:52,180 So this is just a three Russian old truncation of a laboratory fluid. 161 00:17:52,930 --> 00:17:57,280 So you've got three variables X, Y and Z. So these are not space. 162 00:17:57,280 --> 00:18:00,970 These sort of describe a type of circulation. It doesn't really matter. 163 00:18:01,090 --> 00:18:07,059 This this equation has been truncated so much, it now sort of likes to lose contact a bit with the real world. 164 00:18:07,060 --> 00:18:13,930 So it's kind of an idealisation of the real world. So X, Y and Z, just variables thinking the variables. 165 00:18:13,930 --> 00:18:22,150 And in the three equations, the left hand side, the x by d t d y by d t d set by d to use the time rate of change of X, Y, and Z. 166 00:18:22,510 --> 00:18:28,400 And it's given by these terms on the right hand side. The numbers, ten and 28 are not. 167 00:18:28,820 --> 00:18:32,149 They don't have to be precisely those. These are the numbers that Lawrence used. 168 00:18:32,150 --> 00:18:35,540 But you can, you know, choose numbers nearby and you get the same behaviour. 169 00:18:36,840 --> 00:18:44,670 And you notice that in the right hand side you get these terms multiplied together, X and Z multiplied together, X and Y multiplied together. 170 00:18:45,000 --> 00:18:49,590 So it does retain that notion of nonlinearity, which as we come to, is all important. 171 00:18:51,260 --> 00:19:00,380 And when Lawrence integrated these equations. So now we're looking at the the X component variable as a function of time. 172 00:19:01,220 --> 00:19:07,760 You can see it's, it's, it doesn't have any it doesn't you know, it, it looks very irregular. 173 00:19:07,760 --> 00:19:15,200 There are periods when it seems to be up in one state, but then it kind of jumps down or periods when it's down here and then it jumps up. 174 00:19:16,660 --> 00:19:23,980 So this is exactly what he was looking for to try to kind of prove this counterexample to the statistical model. 175 00:19:25,240 --> 00:19:31,149 But the more Lorenz looked at this, the realised he didn't really understand what was going on, 176 00:19:31,150 --> 00:19:34,450 what on earth, you know, what is causing this behaviour. 177 00:19:35,970 --> 00:19:39,840 And he had the real insight into plotting it in a different way. 178 00:19:41,550 --> 00:19:44,129 So instead of taking one variable and plotting it again, 179 00:19:44,130 --> 00:19:50,010 this time you take all three variables and imagine a three dimensional space span by those three 180 00:19:50,010 --> 00:19:57,180 variables and then let time kind of represent be represented by the length along a trajectory. 181 00:19:57,180 --> 00:20:04,230 So this represents some fairly random starting condition in this three dimensional space of X, Y and Z. 182 00:20:04,290 --> 00:20:08,430 So you give it anything you like and then let it go with his computer. 183 00:20:08,970 --> 00:20:13,230 And what he found was that the trajectory start to fall. 184 00:20:15,540 --> 00:20:22,919 Sort of strange shape. Now, the first thing is you can see it's got this these two kind of wings, if you like. 185 00:20:22,920 --> 00:20:27,540 And sometimes people call this a it looks like a butterfly, but that's entirely coincidental. 186 00:20:27,840 --> 00:20:35,219 But it has these two kind of wings which which in a sense describe those that two that kind of regime behaviour, 187 00:20:35,220 --> 00:20:38,550 it tends to be the variable tends to be up here or down here. 188 00:20:38,580 --> 00:20:44,640 Well those if you like up here, up here is, is going around here and down here goes round here. 189 00:20:45,930 --> 00:20:47,729 So that was kind of that was an interesting thing. 190 00:20:47,730 --> 00:20:55,500 But then Laurence sort of said, well, what exactly is this thing that this this trajectory is kind of oscillating on? 191 00:20:55,500 --> 00:21:01,020 What is what is the geometric object behind this? And for a while he thought maybe it's some kind of surface. 192 00:21:01,020 --> 00:21:05,850 These two lobes are lying on the surface and somehow the surface is just sort of glued together. 193 00:21:06,510 --> 00:21:13,060 But then he realised that couldn't be the case because. Then the trajectories would have to kind of cross over each other. 194 00:21:13,100 --> 00:21:19,839 It just didn't work. And it was he agonised about this for by the way, I just want to talk about that. 195 00:21:19,840 --> 00:21:27,290 I've got a. A nice animation of the of how the actual state is not going. 196 00:21:29,470 --> 00:21:32,920 They just don't work. Yeah. 197 00:21:34,380 --> 00:21:36,450 It doesn't seem to work anymore. Okay. Not to worry. 198 00:21:37,170 --> 00:21:43,530 This would have been an animation showing a the state going round in a very irregular way around what is going to go. 199 00:21:44,550 --> 00:21:48,209 Okay. And you've got different perspectives. That's now the Y in the Z. 200 00:21:48,210 --> 00:21:51,290 I think you had the. The Z and the extraction. 201 00:21:51,290 --> 00:22:00,070 That's the name of the X and Y direction. Then it'll rotate round to the. So Lawrence kept asking, So what on earth is this geometry? 202 00:22:00,080 --> 00:22:04,550 What is it? What is it? Actually, what? What how can I describe this in a kind of mathematical way? 203 00:22:06,020 --> 00:22:09,140 And that's when you realise this has to be some kind of fractal. 204 00:22:09,830 --> 00:22:15,889 Now maybe these days we're kind of blasé about fractals, we see them everywhere, but just kind of caution went back to the 1960s. 205 00:22:15,890 --> 00:22:24,680 He's dealing with equations which as I say, Newton would have had no trouble understanding and suddenly how this fractal geometry kind of comes out. 206 00:22:24,690 --> 00:22:30,470 So what I want to do is spend a few minutes trying to get this insight into why why a fractal? 207 00:22:31,670 --> 00:22:36,260 And to do that, I'm actually going to use a slightly simpler dynamical system. 208 00:22:37,430 --> 00:22:48,649 It's just easier to explain. And this actually was was was derived, as you can see, 13 years later, by wrestler in a way, 209 00:22:48,650 --> 00:22:55,220 as a way of of kind of really trying to describe this phenomenon of chaos as as simply as possible. 210 00:22:55,550 --> 00:22:59,630 And again, it's a three component system of differential equation. 211 00:22:59,640 --> 00:23:04,250 So the DOT stands for D by d t, so divided you've actually divided by two. 212 00:23:04,250 --> 00:23:10,370 You've said but slightly different equations. But again, notice the non linearity where pointer is. 213 00:23:11,790 --> 00:23:18,200 Okay. So what I want to do is take you through the rustler a tractor and try to understand why that is fractal. 214 00:23:19,280 --> 00:23:24,200 So I'm going to use some nice pictures from a book by Abraham Henshaw in 1984. 215 00:23:25,280 --> 00:23:31,920 So first let me start with so what I want to introduce are three essential components for this type of chaos. 216 00:23:33,560 --> 00:23:40,310 And the first two components are illustrated here. And this is generic to all chaotic systems. 217 00:23:41,960 --> 00:23:46,570 So imagine you take a little area here. 218 00:23:46,580 --> 00:23:50,450 This is in this space of states, state space, X, Y or Z, whatever. 219 00:23:52,130 --> 00:23:59,720 And we're going to look at, let's say, two, two points on the edge here and let them evolve in time. 220 00:24:00,440 --> 00:24:10,040 Now, what this what what these lines are illustrating is the fact that one of the essential ingredients of chaos is the notion of instability, 221 00:24:10,400 --> 00:24:18,530 that the two points that are initially close start to diverge in general exponentially from each other. 222 00:24:19,560 --> 00:24:27,360 And that's kind of illustrated, as I say, here. So this this point goes off in this direction, whereas this one goes off in this direction. 223 00:24:27,360 --> 00:24:34,380 So this direction here is a kind of direction of instability where initially close trajectories are diverging apart. 224 00:24:34,410 --> 00:24:42,750 So that's one key ingredient. The second key ingredient is actually looking at this in the sort of transverse direction. 225 00:24:44,380 --> 00:24:47,920 So looking at points, let's say, which start on the other edge of the square, 226 00:24:48,520 --> 00:24:53,740 because the second feature of chaos is that these or this type of chaos at least, 227 00:24:54,100 --> 00:24:58,420 is that these trajectories in this direction are actually converging together. 228 00:24:59,680 --> 00:25:04,120 So they're getting closer. Now, the key point is that overall. 229 00:25:05,410 --> 00:25:08,410 Average over the whole system in all time. 230 00:25:08,830 --> 00:25:14,110 It's actually the converging direct directions win over the expanding directions. 231 00:25:15,090 --> 00:25:18,570 So that if I was to measure the area of this. 232 00:25:19,930 --> 00:25:26,640 So here's if I take this area at initial as imagine this is at some initial time and then that evolves into an area. 233 00:25:27,000 --> 00:25:29,910 Imagine looking at this object from the far side. 234 00:25:30,330 --> 00:25:37,950 Then there'd be an area here which was stretched out in this direction but compressed in the transverse direction. 235 00:25:38,340 --> 00:25:43,790 Then on average, the area will be smaller afterwards than it was before. 236 00:25:43,800 --> 00:25:51,110 So this overall shrinkage of area. So these are two key ingredients for chaos. 237 00:25:51,980 --> 00:25:59,030 So this just shows the same thing again, except they've just drawn or they've drawn this the transverse, 238 00:25:59,030 --> 00:26:03,230 if you like, direction as a, as a rather small, um, just much smaller. 239 00:26:03,440 --> 00:26:06,410 Okay. So that's fine. Now we have a problem. 240 00:26:06,530 --> 00:26:12,679 I mean, if, as I say, these if you just imagine exponentials of divergence, then this would just carry on. 241 00:26:12,680 --> 00:26:17,600 They would diverge as far apart as you like, go off to infinity, as it were. 242 00:26:17,990 --> 00:26:26,600 So this can't possibly explain the fact that we have a a kind of a geometry which seems to sit in a in a finite, compact region of state space. 243 00:26:27,200 --> 00:26:30,320 So how do we stop the trajectories going off to infinity? 244 00:26:31,310 --> 00:26:34,880 Well, this is where the third ingredient of nonlinearity comes in. 245 00:26:35,870 --> 00:26:40,999 And in the case of the Russell Attractor, it's rather simple. The nonlinearity just folds one of this. 246 00:26:41,000 --> 00:26:45,450 The surface over like this. So and so. 247 00:26:45,530 --> 00:26:53,390 Just take one hand and just fold it over. So let's have a look at that in a bit more detail. 248 00:26:53,630 --> 00:26:59,160 So let's let's take our. Let's talk a little surface. 249 00:26:59,700 --> 00:27:03,150 Here are two points, two trajectories. 250 00:27:04,290 --> 00:27:10,230 And they kind of evolve and the system becomes nonlinear and the surface folds over like this. 251 00:27:10,740 --> 00:27:17,280 So that one trajectory has gone to the top part and the other trajectory to the to the bottom part there. 252 00:27:17,280 --> 00:27:19,140 And it's kind of folded over in half. 253 00:27:21,230 --> 00:27:31,430 Now, if you remember what I said, if you try to let's try to kind of compress that down into the same cross-sectional area as we started with. 254 00:27:32,740 --> 00:27:41,110 Then what I said, if you if you recall, is that this is the combined area or the total area of this top part. 255 00:27:41,500 --> 00:27:44,590 Plus the bottom part is actually less. 256 00:27:44,920 --> 00:27:48,840 It's smaller because of the overall effect of this contracting this. 257 00:27:48,970 --> 00:27:56,140 This is the effect of dissipation, effectively. This total area is smaller than the the one I started with. 258 00:27:56,710 --> 00:28:05,650 So if I try to compress the whole thing into the same area, there must be a gap between these the top sheet, if you like, and the bottom sheet. 259 00:28:06,010 --> 00:28:10,060 This is a really crucially important point. There's a gap. There has to be a gap. 260 00:28:10,480 --> 00:28:14,320 They don't kind of completely because you've got some compression effectively. 261 00:28:14,530 --> 00:28:17,720 So. All right. So let's take this thing again. 262 00:28:18,670 --> 00:28:22,960 You can imagine this is just like here, but with the gap. But I've sort of blown this up a bit. 263 00:28:23,980 --> 00:28:29,290 And so this kind of comes round as two sheets and we'll go we'll go through the whole exercise again. 264 00:28:30,430 --> 00:28:34,960 Fold it over. And now you see what was two sheets have become force sheets. 265 00:28:35,830 --> 00:28:39,310 Now this gap. 266 00:28:40,410 --> 00:28:45,390 Has now kind of formed two small gaps, if you like, here and here. 267 00:28:45,690 --> 00:28:51,030 And a bigger gap has appeared in between them for the same reason that I said before. 268 00:28:52,400 --> 00:28:59,540 And now we do it again. We got now these four sheets we folded over and now we've got eight sheets with a big gap, 269 00:28:59,960 --> 00:29:05,990 two small gaps and four really small gaps, which was the original gap done, done twice over. 270 00:29:07,160 --> 00:29:13,010 Now, if we keep doing this over and over and over and over and over and over and over and over, what are we going to end up with? 271 00:29:14,620 --> 00:29:23,380 Okay. Well, one way to to try to understand that is to take a cross-section through this this structure here. 272 00:29:23,770 --> 00:29:30,759 After, you know, a very large number of of these types of iterations, this is sometimes actually called a Lorenz section. 273 00:29:30,760 --> 00:29:36,790 The whole thing is sometimes called a Poincaré section. And this kind of transverse through the trajectory is called a Lorenz section. 274 00:29:37,930 --> 00:29:44,889 So what is this? Well, this was this structure was discovered some years earlier, in fact, 275 00:29:44,890 --> 00:29:51,070 in the late 19th century, early 20th century by the great mathematician George Cantor. 276 00:29:51,880 --> 00:29:58,450 And his reason for for this he invented this set called the Cantor set was really nothing to do with chaos per say, 277 00:29:58,450 --> 00:30:02,390 but really to try to understand a bit more about the nature of numbers. 278 00:30:02,550 --> 00:30:10,570 And this is an interesting set is an interesting example of a of a set which has an uncountable infinite number of points. 279 00:30:10,570 --> 00:30:14,920 But but the points actually take up no volume at all. 280 00:30:14,920 --> 00:30:18,130 No volume at all in the on the on the line. 281 00:30:19,060 --> 00:30:29,680 So the Cantor set is is constructed by starting with just, let's say, the line between zero and one and throw away the middle third. 282 00:30:30,130 --> 00:30:34,690 So you got two pieces and then throw away the middle third of the remaining two pieces 283 00:30:35,080 --> 00:30:38,440 and then throw away the middle third of the remaining four pieces and continue, 284 00:30:38,440 --> 00:30:46,000 continue. And then just take the intersection of all of these iterates and you're left with this set. 285 00:30:46,360 --> 00:30:51,000 Now, so this big the big gap. Is is if you like. 286 00:30:51,300 --> 00:30:57,300 Is this part of the of the rustler a attractor or it's a gap in the rustler tractor. 287 00:30:57,660 --> 00:31:03,000 And then the next iterate are these two smaller gaps and so on and so forth. 288 00:31:03,030 --> 00:31:07,559 So we have this. So this is where fractals basically come from. 289 00:31:07,560 --> 00:31:11,940 The three ingredients, instability, dissipation, non linearity. 290 00:31:14,220 --> 00:31:22,740 And this is what Lawrence said in his 63 paper. We see that each surface is really a pair of surfaces, so that where they're they appear to merge. 291 00:31:24,630 --> 00:31:25,860 They're really four surfaces. 292 00:31:25,860 --> 00:31:32,159 Continuing this process for another circuit, we see they're eight processes and we finally conclude there's an infinite complex of surfaces, 293 00:31:32,160 --> 00:31:38,160 each extremely close to one another or other, the two merging surfaces, I think. 294 00:31:38,190 --> 00:31:41,850 Ian Stewart in his popular science book, Does God Play Dice? 295 00:31:42,270 --> 00:31:49,979 Absolutely Nails the reaction, I think of a lot of mathematicians when they read this paper and he said, 296 00:31:49,980 --> 00:31:54,780 when I read Lawrence's words, I get a prickling at the back of my neck and my hair stands on end. 297 00:31:54,810 --> 00:32:00,970 He knew 34 years ago. He knew. And when I look more closely, I'm even more impressed in a mere 12 pages. 298 00:32:00,990 --> 00:32:05,370 Lawrence anticipated several major ideas of nonlinear dynamics before it became fashionable. 299 00:32:06,930 --> 00:32:15,360 So if I had to say, you know, one thing that Lawrence for me absolutely characterises the genius of Lawrence, 300 00:32:15,990 --> 00:32:20,430 it's actually not so much the, you know, this divergence of trajectories, 301 00:32:20,730 --> 00:32:27,540 but the realisation of this fractal geometry, which underpins these sets of differential equations, 302 00:32:27,540 --> 00:32:31,290 which by all accounts should look very smooth and unexceptional. 303 00:32:31,470 --> 00:32:35,629 Unexceptional. So with that in mind, 304 00:32:35,630 --> 00:32:43,280 I am going to make the claim that in some sense Lawrence provides a bridge really between the classical 305 00:32:43,850 --> 00:32:51,930 physics and classical mathematics of Newton and some of the real key ideas in 20th century math. 306 00:32:51,950 --> 00:32:57,860 So we've already mentioned Cantor. This is Kurt Gödel and this is Alan Turing. 307 00:32:58,310 --> 00:33:10,400 And I'm sure people know that some. They really revolutionised our ideas in mathematics by bi by raising the notion that there are there 308 00:33:10,400 --> 00:33:18,620 are things which are true in mathematics but not provably true or in Turing's language propositions, 309 00:33:18,620 --> 00:33:24,620 which are kind of undecidable algorithmically, no matter how complex or your computation might be, 310 00:33:24,980 --> 00:33:29,090 you just cannot establish the truth of of certain propositions. 311 00:33:29,600 --> 00:33:37,549 And it turns out that many of the properties of these fractal geometries are quite isa morphic, 312 00:33:37,550 --> 00:33:46,010 quite similar to the types of propositions of good Alan Turing that are undecidable. 313 00:33:47,140 --> 00:33:50,920 In fact, the papers written on this showing this in a formal way. 314 00:33:51,850 --> 00:33:56,379 So there is this bridge between, as I say, the calculus of Newton, which, as I say, 315 00:33:56,380 --> 00:33:59,530 Newton would look at these equations, say, yes, I take that, I understand that. 316 00:34:00,880 --> 00:34:08,680 And but through this remarkable geometry that they generate, we have these links to some of the most profound things in in mathematics. 317 00:34:08,860 --> 00:34:14,319 Now, I've included Andrew Wiles here because there's lots of other interesting things you can ask. 318 00:34:14,320 --> 00:34:23,530 For example, suppose I want to have a type of arithmetic where I add numbers on on this or on this fractal set, on this cantor set, 319 00:34:23,530 --> 00:34:28,359 let's say on this transverse cantor set of multiply numbers such that the sum 320 00:34:28,360 --> 00:34:32,350 of the numbers or the product of the numbers also lies on the cantor set. 321 00:34:32,440 --> 00:34:41,710 Now, can we do arithmetic on the Cantor sets? It turns out you need a special type of number system to do that called padding numbers. 322 00:34:42,670 --> 00:34:50,230 So I think numbers are a very rigorous branch of pure mathematics that has a very close link to fractal geometry. 323 00:34:50,950 --> 00:34:58,300 And as I say, they they respect the sort of geometric constraints of fractals in doing algebra. 324 00:34:59,140 --> 00:35:05,020 And it turns out that a lot of the deep theory a number theorems from poetic numbers 325 00:35:05,020 --> 00:35:11,380 systems were very important in Andrew's celebrated proof of Fermat's Last Theorem. 326 00:35:13,350 --> 00:35:24,959 One of the things which really intrigues me as a physicist is whether there are links between these ideas and quantum physics. 327 00:35:24,960 --> 00:35:35,280 So people may recognise Erwin Schrödinger and Heisenberg, Paul Dirac and another very celebrated Oxford mathematician, Roger Penrose. 328 00:35:37,680 --> 00:35:45,540 Suddenly three of these people, Schrödinger, Dirac and Penrose, I'm not so sure about Heisenberg, 329 00:35:45,540 --> 00:35:49,630 but the three of them were very uncomfortable with quantum theory. 330 00:35:49,650 --> 00:35:56,100 It works and it does work to this day, but the foundations are very the why this keeps jumping. 331 00:35:56,550 --> 00:36:00,510 The foundations are very kind of bizarre and difficult to understand, 332 00:36:00,510 --> 00:36:09,630 and I think all three of them felt that there must be something deeper underpinning quantum theory, something much more deterministic. 333 00:36:10,170 --> 00:36:16,379 But Roger Penrose in his books like The Emperor's New Mind, which I'm sure some of you will have read very much, 334 00:36:16,380 --> 00:36:23,670 emphasised that if we if there is some deterministic underpinning to quantum theory, it has to have some notion of non compute ability. 335 00:36:23,670 --> 00:36:27,210 This undesired ability has to feature somewhere deeply. 336 00:36:27,510 --> 00:36:31,710 Just an ordinary type of deterministic system won't do it. 337 00:36:32,400 --> 00:36:35,790 So the reason I've included Roger here is because he has very much stressed the 338 00:36:35,790 --> 00:36:41,250 potential role of non compute ability in going deeper into fundamental physics. 339 00:36:41,640 --> 00:36:46,170 So again, perhaps these types of fractal objects play a role. 340 00:36:46,170 --> 00:36:54,270 I personally believe they do, but one has to say at the moment it's a matter for uh, for study and so on. 341 00:36:54,270 --> 00:36:59,910 So I'm not going to dwell upon this any more in the talk, but I'm happy to talk to people afterwards if they're interested. 342 00:37:01,200 --> 00:37:11,400 Chaos theory, for sure, has revolutionised many different branches of science in physics, in biology, in engineering, in economics and social science. 343 00:37:11,550 --> 00:37:17,430 It's hard to think of any area where it's left untouched. Unfortunately, it's sometimes also abused. 344 00:37:18,910 --> 00:37:22,270 So I want to give an example of an abuse of chaos theory. 345 00:37:22,270 --> 00:37:27,630 And this is actually something that's close to my heart because I have to deal with these sorts of people quite a lot. 346 00:37:27,680 --> 00:37:31,500 The climate, well, they don't like to be called deniers, climate sceptics. 347 00:37:32,830 --> 00:37:38,460 Now, one argument which is quite common actually goes like this You guys, okay, 348 00:37:38,470 --> 00:37:41,740 you might be able to forecast the weather tomorrow, but you're pretty hopeless. 349 00:37:42,160 --> 00:37:47,980 And even then, you sometimes get it wrong. And, you know, a couple of weeks ahead is pretty dodgy. 350 00:37:48,400 --> 00:37:52,120 A year ahead, no chance. So what wasn't on it? 351 00:37:52,130 --> 00:37:56,680 Why on earth should I believe what you say about the climate 100 years from now? 352 00:37:57,790 --> 00:38:04,119 So this is this was this was brought up not so long ago in a review of a book in The Telegraph. 353 00:38:04,120 --> 00:38:12,640 Daily Telegraph. The game's up for climate change believers, the theory of global warming as a gigantic weather forecast, 354 00:38:12,970 --> 00:38:17,440 and therefore it can have a century or more and therefore can have no value as a prediction. 355 00:38:17,830 --> 00:38:21,190 Okay. So I just want to debunk that very briefly and then we'll move on. 356 00:38:22,120 --> 00:38:26,770 Because basically that is to misunderstand the problem of climate change. 357 00:38:28,960 --> 00:38:36,280 What I've done here is to do a kind of climate change experiment in Lawrence world by adding an extra term to one of the equations. 358 00:38:37,480 --> 00:38:40,630 Now, just think of this term here. This is actually just going to be a number. 359 00:38:40,990 --> 00:38:47,350 But just think of this as a kind of a surrogate for doubling carbon dioxide concentrations in the atmosphere. 360 00:38:47,350 --> 00:38:52,120 So I'm a sort of applying an extra forcing term, if you like, to the equations. 361 00:38:52,510 --> 00:38:57,669 So how does that affect the Lorenz the Lorenz system? 362 00:38:57,670 --> 00:39:08,709 So here's again the equation X component as a function of time before the system would oscillate roughly between these two kind of regimes, 363 00:39:08,710 --> 00:39:11,980 the two lobes of the butterfly with equal sort of frequency. 364 00:39:11,980 --> 00:39:16,780 So it is likely to be up here in one lobe of the butterfly, as it would be to be down here. 365 00:39:17,680 --> 00:39:22,480 What's happened with this additional term is that I've increase the likelihood of it 366 00:39:22,510 --> 00:39:26,650 being in the upper lobe and decrease the likelihood of it being in this lower lobe. 367 00:39:27,370 --> 00:39:32,979 Now, the system is still chaotic because if I want to predict in detail at one of the trajectories, 368 00:39:32,980 --> 00:39:38,500 if I want to make a weather forecast, if you like, in this Lorenz world, it's still going to be predictable. 369 00:39:38,500 --> 00:39:42,340 It's still going to be subject to uncertainties in the initial conditions. 370 00:39:42,640 --> 00:39:47,980 But what is predictable is this kind of gross statistic that the probability. 371 00:39:49,010 --> 00:39:53,270 Of the state being up here has been affected in a very predictable way. 372 00:39:53,750 --> 00:40:00,530 In fact, we can you know, we can actually plot this on in the Lawrence attractor world in the state space. 373 00:40:00,950 --> 00:40:08,179 So here's the original Lawrence Attractor in the new system, these two kind of what we call centroid. 374 00:40:08,180 --> 00:40:10,790 So right in pretty much the same place as they were before. 375 00:40:11,090 --> 00:40:16,550 But now the system spends a lot more time zipping around here than it does zipping around here. 376 00:40:18,530 --> 00:40:23,060 So we could view in this perspective, we could view climate change as a problem in geometry. 377 00:40:24,100 --> 00:40:33,640 The question is how is the geometry of this attractor being affected by the addition of this term, which I've now set equal to ten? 378 00:40:35,470 --> 00:40:39,590 So it's important to understand this. So climate change, this is the problem of climate change. 379 00:40:39,610 --> 00:40:48,099 How does the whole climate attractor it, which basically means the statistics of weather change as we double carbon dioxide levels, 380 00:40:48,100 --> 00:40:53,890 which we surely will be sometime later this century or so over pre-industrial. 381 00:40:54,640 --> 00:40:58,210 How how how does that change the statistics of weather? 382 00:40:58,630 --> 00:41:02,230 And that's not the same problem at all as completely different class of problem to 383 00:41:02,230 --> 00:41:07,780 saying how does a particular trajectory evolve on its on the climate attractor. 384 00:41:08,770 --> 00:41:12,820 Over some from some initial condition. Okay. 385 00:41:13,060 --> 00:41:24,970 Uh, let me just now move on to the sort of part of the talk I wanted to, uh, I wanted to get to, which is what did Laurence actually mean? 386 00:41:27,430 --> 00:41:31,540 By the butterfly effect. And you'll see that, in fact, he wasn't talking about his 63 paper at all. 387 00:41:33,900 --> 00:41:37,610 He's talking about the fact that the weather is a multi scale system. 388 00:41:37,620 --> 00:41:48,299 So we have we have a you know, if you imagine a large low pressure system trundling across the Atlantic embedded in that low pressure system, 389 00:41:48,300 --> 00:41:54,030 there'll be, let's say, you know, thunderstorm clouds, big thunderstorm clouds where they match the some of the storm clouds, 390 00:41:54,210 --> 00:41:58,560 maybe 100 kilometres in scale, whereas the weather system itself is a thousand kilometres in scale. 391 00:42:00,780 --> 00:42:05,819 Embedded in that thunderstorm cloud, there are small sub clouds, turbulent eddies, 392 00:42:05,820 --> 00:42:10,139 and within those sub cloud, turbulent eddies there, yet smaller turbulence eddies. 393 00:42:10,140 --> 00:42:14,310 So we're looking at this this Russian doll hierarchy again. 394 00:42:16,520 --> 00:42:22,489 Now, Laurence said, okay, let's suppose our problem is to predict this weather pattern, the large scale weather pattern. 395 00:42:22,490 --> 00:42:23,630 That's what we're interested in. 396 00:42:25,210 --> 00:42:31,330 We might be able to measure, you know, the initial the starting conditions for this weather pattern very, very accurately. 397 00:42:32,710 --> 00:42:38,770 In fact, we can imagine almost perfectly for the sake of argument, measuring the initial conditions for that weather scale. 398 00:42:39,250 --> 00:42:44,770 But that's not going to give us indefinite predictability because sooner or later the fact that 399 00:42:44,770 --> 00:42:51,520 we haven't been able to measure the the cloud scales or the sub cloud scale motions perfectly, 400 00:42:51,520 --> 00:42:55,989 that's going to catch up with us because these areas are going to kind of propagate up non-linear, 401 00:42:55,990 --> 00:42:59,260 only from the small scale to the larger scale to the very large scale. 402 00:43:01,210 --> 00:43:06,550 If you want to actually read this, he's Lawrence himself wrote a popular book called The Essence of Chaos. 403 00:43:06,550 --> 00:43:14,410 And in the appendix, he actually describes what went into this talk in the triple OAS meeting in 1972, 404 00:43:15,130 --> 00:43:18,340 and he talks about errors in the course of scrapped structure. 405 00:43:18,350 --> 00:43:23,980 So that's a large scale weather pattern and then errors in these final structures, which are the clouds. 406 00:43:24,250 --> 00:43:29,620 And then in fact, the fact that errors in the final structure can start to induce errors in the course of structure. 407 00:43:31,170 --> 00:43:35,639 What I've decided to do for this talk is actually not to go through in detail with what he wrote, 408 00:43:35,640 --> 00:43:43,320 but return to my Russian doll example and try to describe what he what his thinking was with that. 409 00:43:44,810 --> 00:43:49,879 So let's imagine the Russian doll, the big Russian doll is this big low pressure system trundling across the Atlantic. 410 00:43:49,880 --> 00:43:53,600 And that's the thing we want to predict. We want to predict that as far ahead as we possibly can. 411 00:43:54,620 --> 00:44:01,280 And let's imagine we've got some observing, some instrument which has no error at all. 412 00:44:01,310 --> 00:44:10,110 I mean, infinitesimal, just quantum mechanics, just for the sake of argument, measures perfectly measures whatever its wants to do. 413 00:44:10,130 --> 00:44:17,810 Temperature, pressure and stuff like that. Now we have a whole load of these instruments and we're going to adopt them around the whole globe. 414 00:44:17,840 --> 00:44:24,980 We've got enough of them. We can dump them around the whole globe in a regular network where the distance between the two, 415 00:44:25,040 --> 00:44:28,970 between any two of these measuring systems is sufficiently, 416 00:44:29,840 --> 00:44:36,530 let's say, small, that it can it can resolve, you know, this this low pressure system really, really well. 417 00:44:38,470 --> 00:44:46,990 So that the initial conditions for this low for the scale, this low pressure system, the scale of this low pressure system are known almost perfectly. 418 00:44:48,220 --> 00:44:57,640 But the distance between these measuring objects is not sufficiently small that I can measure the initial conditions for all these smaller eddies, 419 00:44:57,640 --> 00:45:03,209 the smaller wells of Richardson. So I'm going to imagine that with that information. 420 00:45:03,210 --> 00:45:08,390 Let's say I can predict ahead five days. I mean, it's a reasonable sort of number. 421 00:45:09,740 --> 00:45:15,320 So what Lawrence said was, well, maybe we're not happy with that. How would we extend that predictions? 422 00:45:15,500 --> 00:45:17,390 Horizon We want to predict further. 423 00:45:18,020 --> 00:45:23,720 Well, what's limiting us in this case is not our ability to measure this scale, because we've measured this perfectly. 424 00:45:24,170 --> 00:45:28,250 What's limiting it is the fact that we don't have any information about this scale. 425 00:45:28,970 --> 00:45:34,760 So we'll stick a whole load more of these instruments down, filling the gaps between the ones we had. 426 00:45:35,330 --> 00:45:40,370 So now we have enough resolution that we can measure the initial conditions of this scale here. 427 00:45:40,940 --> 00:45:46,280 So the question is, how much is that going to bias? How much extra prediction skill is that going to give us? 428 00:45:47,360 --> 00:45:52,280 Now, Lawrence says the answer is not ten days, because these these things, 429 00:45:52,730 --> 00:45:56,960 the errors are likely to be growing faster for these smaller scale features. 430 00:45:57,620 --> 00:46:00,019 So, for example, you mentioned the cloud system, the error, 431 00:46:00,020 --> 00:46:05,660 the time it takes because the circulation patterns are more vigorous in a in a in an individual cloud. 432 00:46:06,110 --> 00:46:11,329 If you have a flown through a cloud in an aeroplane, you know, speaking of thunderstorm cloud, there's a very vigorous circulation. 433 00:46:11,330 --> 00:46:19,080 So air is a growing much more rapidly. So let's I'm kind of simplifying the for the experts. 434 00:46:19,090 --> 00:46:24,960 I'm simplifying things a little bit just to make the point. So but nevertheless, the idea is, is qualitatively correct. 435 00:46:24,990 --> 00:46:31,080 So let's imagine for this second, you only get two and a half days of extra prediction skill. 436 00:46:31,090 --> 00:46:39,000 So you've gone up now from five days to seven one half days. So Lauretta says, okay, that's fine, but let's, let's say we want to go further. 437 00:46:39,010 --> 00:46:45,930 So let's, let's have a whole load. More measurement systems are now fill in even the gaps in those gaps. 438 00:46:46,260 --> 00:46:52,470 So then now we can even measure this Russian doll initially perfectly and he says, okay, 439 00:46:52,980 --> 00:46:58,350 again, we'll only because these are growing yet faster, the errors are growing faster. 440 00:46:58,770 --> 00:47:03,240 We may only get an extra C I'm I'm halving the extra time each step. 441 00:47:03,720 --> 00:47:12,330 So now we only get an extra day and a quarter. So you can see this continual investment in imperfect measurement measurement systems, 442 00:47:13,200 --> 00:47:20,250 which is driving the initial error closer and closer to zero, is actually buying us less and less prediction time. 443 00:47:21,870 --> 00:47:27,839 And what Lawrence says. Let's take this to the limit and imagine we've got an infinite number of measuring 444 00:47:27,840 --> 00:47:32,490 systems which can basically take us down to infinitesimally small scales. 445 00:47:33,270 --> 00:47:43,710 He's saying that what this will bias in terms of prediction time is a series which if with these particular choices of doubling times, 446 00:47:44,100 --> 00:47:47,130 will actually converge to a finite number. 447 00:47:47,940 --> 00:47:52,650 Now, normally we have when we have series, we think of convergence as a good thing and divergence is a bad thing. 448 00:47:52,950 --> 00:48:02,040 But here, convergence is a bad thing because it's really limiting your prediction capability to some finite time, 449 00:48:02,040 --> 00:48:05,670 which in this simple calculation turns out to be ten days. 450 00:48:06,570 --> 00:48:10,500 So this has a remarkable. If you think about it, this is a remarkable thing. 451 00:48:10,500 --> 00:48:19,350 If it's correct, it says. No matter how small my initial error is, I will never be able to predict more than ten days ahead. 452 00:48:20,490 --> 00:48:29,370 Now, just think that's completely different to the 1963 paper, because the 63 paper is about just these three equations, 453 00:48:29,370 --> 00:48:36,030 which, although it's very difficult to predict them, you can always in principle predict as far as ahead as you like. 454 00:48:36,330 --> 00:48:41,740 Providing your initial error is sufficiently small. But here's a system where that's not true. 455 00:48:43,600 --> 00:48:51,010 And this is what it's about in his 69 paper, which appeared in a Swedish journal called Tell US. 456 00:48:51,100 --> 00:48:59,260 And if his 63 paper was, you know, is considered to be in an obscure journal, this is obscure squared, I'm afraid. 457 00:48:59,320 --> 00:49:02,610 I'm afraid so. Very, very not. 458 00:49:02,620 --> 00:49:10,509 Not really. Well read at all. But it's worth just if you don't mind just reading, I'll read out the the part of the abstract. 459 00:49:10,510 --> 00:49:15,190 So it is because it's really important this he claims it is he proposed that it is proposed 460 00:49:15,190 --> 00:49:19,630 that certain formally deterministic fluid systems which possess many scales of motion. 461 00:49:19,670 --> 00:49:26,140 That's the key point. Many scales of motion, not the three rationales, but a kind of essentially an infinity of Russian goals, 462 00:49:27,040 --> 00:49:30,370 are observationally indistinguishable for in deterministic systems. 463 00:49:30,730 --> 00:49:35,140 Specifically, that two states of the system, differing initially by a small observational error, 464 00:49:35,650 --> 00:49:42,129 will evolve into two states differing as greatly as randomly chosen states of the system with a within a finite time interval. 465 00:49:42,130 --> 00:49:50,260 In our calculation, it was ten days. Now, here's the real Ryder kicker, which cannot be lengthened by reducing the amplitude of the initial error. 466 00:49:51,400 --> 00:49:59,020 So this is very radical. So what I'm proposing to do here today is to illustrate so as to introduce you to two butterflies. 467 00:49:59,340 --> 00:50:04,450 Now it's we're never going to stop the butterfly effect being used to describe low order chaos. 468 00:50:04,900 --> 00:50:10,920 But I'm going to call this the common butterfly effect. So here's a common blue on the common butterfly. 469 00:50:11,090 --> 00:50:18,250 This is the one that everybody thinks of when they talk about the butterfly effect is about sensitive dependence on initial conditions. 470 00:50:18,820 --> 00:50:23,920 So that means it's certainly difficult to predict the future, but it's not impossible in principle. 471 00:50:23,920 --> 00:50:28,930 You can predict in something like a Lorenz 63 system, you can predict as far ahead as you like. 472 00:50:29,200 --> 00:50:36,450 Providing the initial error is small enough. On the other hand, I'm going to use this monarch butterfly. 473 00:50:36,460 --> 00:50:43,960 So this is a slight, a weak pun on the fact that this is monarch is a royal member of royalty and royal reality in Spanish or something is royal. 474 00:50:43,970 --> 00:50:47,050 So the real butterfly effect, the royal butterfly effect, if you like, 475 00:50:47,770 --> 00:50:52,419 is the 69 paper which basically says that there are finite predictability 476 00:50:52,420 --> 00:50:56,320 horizons which cannot be extended by reducing uncertainty in initial conditions. 477 00:50:58,400 --> 00:51:02,600 So we know that this is this has been certainly well verified. 478 00:51:02,600 --> 00:51:07,130 So the common butterfly effect is been is kind of absolutely part of standard science. 479 00:51:07,370 --> 00:51:12,500 How about this real butterfly effect? Is this part of a of a kind of rigorous science? 480 00:51:14,180 --> 00:51:18,829 Let's just put this into a slightly more mathematical phraseology, 481 00:51:18,830 --> 00:51:28,400 because what what we're saying here is if you take the initial conditions and change them as slightly as you like in a finite time, 482 00:51:28,430 --> 00:51:31,850 you'll you'll diverge, too, to find actually different solutions. 483 00:51:33,800 --> 00:51:42,680 Now in math, in mathematics, problems are referred to as well posed or ill posed according to certain properties that they have, 484 00:51:43,520 --> 00:51:46,010 apparently from a definition given originally by had them. 485 00:51:47,450 --> 00:51:54,140 So problems are well posed if solutions exist, they're unique and the solutions depend continuously on the data. 486 00:51:54,170 --> 00:51:55,790 This means initial data in our case. 487 00:51:56,180 --> 00:52:04,730 Now, this continuously means if you change the initial conditions very, very slightly, you're just going to change the solution slightly. 488 00:52:05,480 --> 00:52:14,840 And that's true in the 63 chaos, but it's not true in the 69 case, in the 69 real chaos, real butterfly effect. 489 00:52:15,920 --> 00:52:22,190 So if this effect is really true, it means that the navier-stokes initial value problem is not well posed in this sense. 490 00:52:22,850 --> 00:52:28,880 So now this has becomes a problem in mathematics. It's the navier-stokes problem for three dimensional, turbulent fluid. 491 00:52:29,900 --> 00:52:31,670 The initial value problem well posed. 492 00:52:33,170 --> 00:52:39,200 And this is why I say this is actually turns out to be one of the great unsolved problems in 21st century mathematics. 493 00:52:39,620 --> 00:52:49,010 So as I'm sure many of you know, the Clay Mathematics Institute put out these kind of key problems, unsolved problems in mathematics, 494 00:52:50,420 --> 00:52:55,580 which include famous things like this is the beginning of this millennium, the Riemann hypothesis and so on. 495 00:52:55,610 --> 00:52:57,590 And here's the the Navier-stokes one. 496 00:52:57,890 --> 00:53:04,219 It's actually not framed in terms of this continuity property, but rather whether solutions exist and are unique. 497 00:53:04,220 --> 00:53:09,290 And even that's not known. And we need to know that even before we can understand this property of continuity. 498 00:53:10,250 --> 00:53:15,680 So I think it's fascinating to me that what Lawrence speculated about in 69 and 499 00:53:15,680 --> 00:53:21,110 then this triple as talk in 72 actually refers to a very deep problem in maths. 500 00:53:21,770 --> 00:53:31,400 So if I got just a few minutes to finish, okay, so I want to kind of get back to two more practical issues because a question you might ask is, 501 00:53:31,640 --> 00:53:36,350 is it really the case then that we're limited in weather prediction to ten days 502 00:53:36,350 --> 00:53:40,880 or something and things aren't quite as simple as that in practice because. 503 00:53:43,210 --> 00:53:50,020 Ten days may well be sort of roughly an average timescale that we can make sort of detailed predictions about the weather. 504 00:53:50,380 --> 00:53:56,350 But what we rapidly find out is that there are many situations where we can make much longer range predictions, 505 00:53:56,350 --> 00:54:03,790 apparently with with quite good skill, but equally other situations where predictions do in fact go wrong within a couple of days. 506 00:54:05,200 --> 00:54:14,010 And that's very well illustrated going back to the 63 paper and actually looking at how errors, if you like, or uncertainty. 507 00:54:14,020 --> 00:54:19,360 So imagine this is an initial state with some kind of ball of uncertainty associated with it. 508 00:54:20,140 --> 00:54:28,660 This actually shows that there are some parts of the Lorenz attractor that are extremely stable, so small perturbations hardly grow at all. 509 00:54:28,930 --> 00:54:33,730 So this is telling you there are some parts of the system which are actually very predictable, 510 00:54:34,450 --> 00:54:40,450 other parts of the situation, other parts of the attractor, where you start to explore the state space down here, 511 00:54:40,720 --> 00:54:47,290 where you do start to see unpredictability and some initial conditions where the uncertainty just explodes. 512 00:54:49,720 --> 00:54:53,980 Now, those of you, most of you in the audience are not old enough. 513 00:54:53,980 --> 00:54:57,129 But I know there are a few people to remember this. 514 00:54:57,130 --> 00:55:06,590 But one of the most iconic weather events of the last maybe 300 years occurred almost 30 years ago, October 1987. 515 00:55:07,270 --> 00:55:14,620 And this poor guy whose name is Michael Fish, although he's I have to say, he makes a great after dinner living out of speech, 516 00:55:14,620 --> 00:55:20,859 living from this famously made probably the worst weather forecast in history by 517 00:55:20,860 --> 00:55:25,540 saying there's no chance of a hurricane hitting the UK in the next a day or so. 518 00:55:25,900 --> 00:55:39,860 And indeed, a hurricane did did hit. And this is actually a great example of this very intermittent phenomenon of explosive unpredictability. 519 00:55:39,890 --> 00:55:49,440 So what I'm going to do here is just animate two weather forecasts from two almost identical initial conditions. 520 00:55:49,460 --> 00:55:52,880 These are these are surface pressure maps, almost identical. 521 00:55:52,890 --> 00:56:00,280 Not quite identical. You can look in detail. You'll see differences. With just a couple of days before the storm hit. 522 00:56:00,400 --> 00:56:08,560 So just run this forward and they start off tracking each other pretty well. 523 00:56:08,830 --> 00:56:17,170 But then leading up to the fateful morning of the 16th, I think it was, you see that these are really quite different. 524 00:56:17,440 --> 00:56:23,350 This would be a very kind of benign day, and this is a really intense vortex. 525 00:56:24,310 --> 00:56:31,000 Poor Michael Fish was given essentially this solution by the Met Office, but for a flap of a butterfly's wing, 526 00:56:31,240 --> 00:56:35,910 he would have been given this one and he would then have been a national hero instead. 527 00:56:36,880 --> 00:56:41,500 Actually, you shouldn't feel too bad about this because the Met Office did really well out of this poor forecast. 528 00:56:41,500 --> 00:56:48,100 They said, Oh, well, our computers aren't big enough, you need to invest lots more money. And so I don't feel sorry for them. 529 00:56:48,790 --> 00:56:53,319 Okay. So the question is, how would a modern day Michael Fish deal with this situation? 530 00:56:53,320 --> 00:56:58,059 Because chaos is chaos. I mean, chaos in 1987 will still happen again. 531 00:56:58,060 --> 00:57:01,930 So this intermittent explosive unpredictability could still happen in 2017. 532 00:57:02,560 --> 00:57:06,010 So how would a modern day Michael Fish deal with this situation? 533 00:57:06,610 --> 00:57:11,440 Well, these days, with much bigger computers and in particular with massively parallel computers, 534 00:57:12,040 --> 00:57:18,550 we can take advantage of that type of architecture to run weather forecasts in what I call ensemble mode. 535 00:57:19,090 --> 00:57:25,180 So we actually we actually run here 50 different forecasts. 536 00:57:25,180 --> 00:57:28,180 This would be done typically every day at places like the Met Office. 537 00:57:28,660 --> 00:57:34,479 Some are actually everywhere around the globe. Now, you don't run a single weather forecast, you run an ensemble of them and very, 538 00:57:34,480 --> 00:57:42,280 very slightly the initial conditions and look to see ahead of time whether the forecasts are diverging and what you see in this case. 539 00:57:42,280 --> 00:57:45,669 So this is the 87 storm run retrospectively. 540 00:57:45,670 --> 00:57:50,950 It's run with a modern day ensemble system, modern day weather forecast model. 541 00:57:52,120 --> 00:58:03,969 But but on this on this 30 year old case and what this shows at about a two and a half day range is a phenomenal spread in in solutions. 542 00:58:03,970 --> 00:58:09,070 So these are all pressure maps. So you see, you know, you can get pretty much everything from, you know, 543 00:58:09,070 --> 00:58:16,720 absolutely balmy, calm days to to horrendous weather, very, very wide divergence. 544 00:58:17,290 --> 00:58:24,670 Okay. So what would you do with this information? How would you sort of how would you convey this information to the public? 545 00:58:25,360 --> 00:58:34,930 Well, one thing you can do is basically say, okay, how many of these members had hurricane force winds and count them? 546 00:58:36,010 --> 00:58:43,569 And if there were, say, 15 out of the 50 had hurricane force winds, you would say 59, 50, 30 out of 100. 547 00:58:43,570 --> 00:58:46,870 So you say is roughly a 30% chance of. 548 00:58:47,710 --> 00:58:50,850 Hurricane force winds, and that's actually pretty much what we see. 549 00:58:50,860 --> 00:58:59,260 So this is a a contour map showing the probability of hurricane force wind gusts on the 16th of October based on that ensemble forecast. 550 00:58:59,710 --> 00:59:03,730 And these are blue lines here. And the English Channel are up around 30, 40%. 551 00:59:05,720 --> 00:59:09,950 Now. So what do you do with that information? That's up to you. 552 00:59:10,220 --> 00:59:12,380 That's not up to the meteorologist. That's up to you. 553 00:59:12,710 --> 00:59:18,200 If you've just bought a brand new yacht and you're not terribly good at sailing and you're thinking of crossing the channel, 554 00:59:18,920 --> 00:59:27,140 maybe don't want to do it. If you just bought a brand new Lamborghini and it's sitting underneath a big oak tree, maybe you want to move it. 555 00:59:27,440 --> 00:59:33,470 30 or 40% is probably a big enough probability that that would be a prudent thing to do. 556 00:59:33,860 --> 00:59:40,630 But it raises a very interesting and a whole topic for another talk some time about making decisions under uncertainty. 557 00:59:40,640 --> 00:59:46,970 But the main point is, and indeed much better decisions can be made once you have a quantified quantify uncertainty. 558 00:59:47,240 --> 00:59:51,110 And that's precisely what this does. All right. So I think I'm finished. 559 00:59:51,110 --> 00:59:54,530 Sorry for going slightly over time. I'm going to be another shamed advertisement. 560 00:59:54,530 --> 00:59:57,919 If you're interested in reading any more about anything I've spoken about, 561 00:59:57,920 --> 01:00:03,530 I just want to highlight three papers which I can send you or you can get them on the on the Internet. 562 01:00:05,090 --> 01:00:13,579 Most of this talk is actually based on a paper that I wrote with colleague Andrea stirring from the physics department and Gregory Sagan, 563 01:00:13,580 --> 01:00:17,780 who's here in the Maths Institute, about this butterfly real butterfly effect. 564 01:00:17,780 --> 01:00:25,180 And we go into a lot more detail about the theory of the navier-stokes equations and things like that, but as well as the history behind the you know, 565 01:00:25,190 --> 01:00:34,190 what I spoke about, I also wrote a biographical memoir for at Lawrence, who died a few years ago, who was a former member of the Royal Society. 566 01:00:34,200 --> 01:00:38,440 So there's a lot of the biographical background to Lawrence in that which you can 567 01:00:38,520 --> 01:00:42,450 again get from the Royal Society and if you're interested in this area of this. 568 01:00:42,450 --> 01:00:47,629 So what I find rather fascinating linkage between Lawrence and the girdle incompleteness 569 01:00:47,630 --> 01:00:51,920 theorems and some of the ideas of Roger Penrose on non compute mobility in physics. 570 01:00:51,920 --> 01:00:59,330 That's in a paper in contemporary physics published by the Institute of Physics, based on a lecture I gave in the physics department a few years back. 571 01:00:59,990 --> 01:01:00,950 Thank you for your attention.