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00:00Oh
00:30Form and chaos, order and disorder, are like rivals competing for supremacy in the vast arena that is our universe.
00:57Nature throws grotesque shapes and turbulent events at us, and yet within them we strive to find evidence of patterns we can classify and understand.
01:08For centuries, man has struggled to impose order on a perversely irregular world, and what he couldn't understand was often ignored.
01:17Now, armed with a new weapon, the computer, man is at last challenging the kingdom of chaos, describing the shapes and forms of a world where the only straight lines are the ones he has introduced.
01:32A new science is emerging from this challenge, a science which promises to describe and explain the infinite complexity of the busy world in which we live.
01:44It's a science which is producing a new geometry that offers an insight into the surprising paradox that there is order, even within disorder.
02:00It's a science which is turning conventional thinking on its head, a science called chaos.
02:09Mercurial and capricious storms flit ghost-like across the earth.
02:26An uncaring of the consequences cause heat waves in Sydney and hurricanes in Kent.
02:32These are the satellite pictures which show the hurricane of October 1987, sweeping north from the Bay of Biscay towards the south coast of Britain.
02:41The aftermath of such an event is a harsh reminder that weather is one of the more unpredictable features of our daily life.
02:49For a hundred years, scientists have struggled to find the rules that govern our weather, hoping that this would enable them to produce accurate long-range forecasts.
03:00We now know that they struggled in vain, that even with today's technology, the task is simply too good.
03:13The European Centre for Medium-Range Weather Forecasts is the most advanced meteorological centre in the world.
03:20Here, they try to make sense of the diversity of our weather.
03:24Every day, a hundred million measurements gathered from around the world are fed into an enormous computer model of the Earth's atmosphere.
03:36What we do is divide the atmosphere up into segments that are about a hundred kilometres by a hundred kilometres in the horizontal and about one kilometre in the vertical.
03:46And it takes about a million of these segments to represent the whole atmosphere.
03:52And every day, we run the model to produce a weather forecast for ten days into the future.
03:59Now, on the present machine, which is essentially one of the most powerful computers in the world,
04:06this takes about three hours, just under three hours of computing time, to produce a ten-day forecast.
04:12Now, to give you some idea of the complexity that's involved with all these interactions going on,
04:19every second of that three-hour integration period, the computer performs about 400 million calculations.
04:29As recently as 20 years ago, this ability to compute 400 million calculations per second was simply inconceivable.
04:39Indeed, in 1959, meteorologist Edward Lorentz was more than satisfied with the 17 calculations per second
04:46that his brand-new computer could perform.
04:49With it, he made the discovery that gave birth to a new science.
04:54The computer that we did our original work on, this YL-McBee LGP30, here with a console and a working portion here,
05:04this was a medium-speed computer in its day. We first got a hold of it in 1959.
05:11It sounds rather crude by modern standards because the computer took ten whole seconds to print out just one line of data.
05:19We had condensed the numbers so it printed things out to three significant places, although they were carried in the computer to six,
05:26but we didn't think we cared about the last places at all.
05:29So I simply typed in what the computer printed out one time and decided to have it printed out in more detail, that is, more frequently.
05:38And I let the thing run while I went out for coffee, and a couple of hours later, when it had simulated a couple of months of data,
05:48I came back and found it was doing something quite different from what it had done before.
05:52It became obvious that what happened was that I hadn't started with the same conditions.
05:57I started with round-off conditions, so that I started with the original conditions, plus extremely small errors,
06:04but in the course of a couple of simulated months, these errors had grown, they doubled about every four days,
06:09that meant in two months they'd doubled fifteen times, or a factor of thirty thousand or so,
06:15and the two solutions were doing something entirely different.
06:18And I have some of the original output here from the computer.
06:23What I've done and did was to print symbols and then go over it in blue pencil, all to illustrate it,
06:29and this is one of the graphs that I obtained.
06:32If you look at a fair stretch of this, you see that there are certain typical things that this particular system of equations does.
06:40It has rather pointed minima, rather extended maxima with little waves in them,
06:47but it doesn't do exactly the same thing twice.
06:49It does one thing, then it does something like it, but something that isn't quite the same, it's a bit different.
06:56The desktop weather from this second, slightly modified computer run appeared tantalizingly familiar.
07:03Lorenz studied the new data, looking for regions where the output appeared to match the first run.
07:09And yet, although many of the peaks and troughs looked similar to the first printout,
07:14when he tried to match them up, the patterns soon drifted apart.
07:19Lorenz's computer had a message for him, and science.
07:23What Lorenz's work tells us is that no matter how complex our models become,
07:29and no matter how good the initial data becomes,
07:33there will be a limit beyond which we cannot provide accurate forecasts of the instantaneous weather.
07:44The discovery has since become known as the butterfly effect,
07:48because in theory, even the tiny disturbance in the air caused by a single butterfly in flight
07:54can tip the balance of our weather and create a chain reaction straddling time and distance,
07:59ultimately altering the course of a tornado on the other side of the world.
08:05It's an effect which has upset quite a few scientific applecars,
08:09since it means that even the tiniest errors of measurement make accurate prediction impossible in many, if not all, physical systems.
08:17Until quite recently, we explained the world using the rigid geometry of the Greeks.
08:27We lived, it seemed, in a clockwork universe governed by immutable laws.
08:34They were the laws of motion first identified by Sir Isaac Newton in the 17th century.
08:40Initially, he described how gravity rules the motion of the planets.
08:44Subsequently, it was believed that all physical systems obeyed these laws,
08:49and that the key to accurate prediction was simply accurate measurement.
08:56In fact, this new natural philosophy seemed to lead inevitably to a conclusion,
09:01first proposed by French mathematician Pierre-Simon Laplace,
09:04that if you could exactly measure the position and motion of every point in the universe,
09:09predicting the rest of eternity would be simply a matter of doing the science.
09:13Towards the end of the last century, Laplace's compatriot, Henri Poincaré,
09:19began to realize that if the universe were driven by clockwork,
09:24then it ticked in a complex and even unpredictable way.
09:28Applying a new geometry to Newton's laws of motion,
09:31he tried to reach a deeper understanding of the dynamics of nature.
09:35But time and again, he came to the edge of a mathematical abyss,
09:39face to face with unpredictability and chaos.
09:43What had happened? Was Newton wrong?
09:46Clearly Newton was right.
09:48We've been using Newton for 300 years to predict the motion of heavenly bodies,
09:53to predict the motion of spaceships, etc.
09:56On the other hand, when you try to fine-tune Newton,
10:02what you find is something that's a little bit different.
10:07Newton says that, yes, if we know the initial behavior,
10:13the initial conditions perfectly, then we can predict what will happen.
10:17The unfortunate problem there is the if.
10:22We don't know anything perfectly in nature.
10:25It's impossible to pin down precisely to 15, 30, 50 million decimal places
10:30the position of a particle in space.
10:34Let me show you what this little machine does.
10:38What we basically have is a little metal ball,
10:43and underneath this plate are a bunch of magnets.
10:47So this little ball can be attracted to various of these magnets.
10:52Now, it's a very simple physical problem to take the ball and let it go and ask what happens.
11:03That's a motion that should be described by Newtonian physics.
11:09Yet here's something that behaves totally chaotically.
11:11Watch how the ball goes.
11:12It bounces back and forth from side to side in a very irregular pattern.
11:18Let me try to start the ball exactly where I started it before, right about there.
11:24If I start the ball from essentially the same position,
11:28you'll notice that after a very short time,
11:31the ball starts to behave in a completely different manner.
11:37It's the butterfly effect again, in an inherently unpredictable system.
11:41In a predictable system, such as when a ball drops onto a flat surface,
11:45the behavior of one ball closely matches the behavior of others dropped from the same height.
11:50There will be small discrepancies, but plotting the path of the first ball
11:54allows you to approximately predict the paths future balls will take and be right.
12:02Bouncing a ball off some round white pegs changes everything.
12:06Now, knowing the path of one ball will not help you to predict what the paths of others would be.
12:11This is the butterfly effect again.
12:13Tiny variations in the paths of the balls are being amplified on every bounce,
12:18making prediction impossible.
12:23This match is igniting a video version of the butterfly effect known as feedback.
12:28This picture is being generated by a camera which is pointing at its own output on a nearby TV screen.
12:37If we just shut off that picture, you see that as the camera looks at the monitor
12:45and doesn't see a picture, it won't reproduce one.
12:49So to start a picture, we should somehow ignite it.
12:54And as we do that here, we just put a little bit of light in the center of the picture,
13:00which is seen by the camera, and then fed back into the monitor,
13:05and the process gets going.
13:08The process of video feedback.
13:10But I guess you would like to know, do we understand the pattern?
13:16Can we describe how the pattern is emerging?
13:19And I think the honest answer to that question is no.
13:23But to convince you how difficult the question is, let's change the setup slightly.
13:29And one way to do this would be, for example, to just rotate the camera slightly around its axis.
13:36And maybe we do this at this point.
13:41And as we do that, you see, by just rotating the camera about a degree or two,
13:47the picture resulting in the feedback system is totally different.
13:53The video camera is scanning the monitor 50 times every second.
13:58But because the camera system is never perfect, tiny distortions to the picture are created by the camera's electronics,
14:06the lenses, and the monitor itself.
14:09Camera distortions, represented here by the letter C, are added to the original picture
14:15to create a new but significantly different picture on the monitor.
14:19The camera sees this new picture and alters it again.
14:22Each frame changes from the preceding one as the system repeats the process over and over.
14:29If C changes, such as by tilting the camera, then the type of image changes.
14:35This process of repetition, or iteration as it's known, is something that computers are very good at doing.
14:41But computers need numbers, not pictures.
14:44So, change the pictures to a symbol, X.
14:48And the mathematical equivalent of feedback might look something like this.
14:53The interesting part is what happens to X when you change C, the equivalent of tilting the camera.
14:59Sometimes the variations in X will smooth themselves out.
15:03And at other values of C, X may seesaw periodically from one point to another.
15:09But at certain other values of C, the resulting X's will bounce chaotically in a patternless sequence.
15:16One of the surprises of chaos is that all these different behaviours can be displayed in just one system.
15:25This computer simulation shows the behaviour of a particular kind of pendulum, a double pendulum.
15:33Sometimes it will follow a regular path which it will repeat forever.
15:38It's a prisoner on that path, tracing out what is known to mathematicians as its attractor.
15:44But nudge slightly harder, a subtle change occurs in the motion.
15:48It still looks periodic and regular, but it's not.
15:52It never actually repeats itself.
15:55The pendulum has started on the road to chaos.
15:58And under these new conditions, it's now drawn to a different attractor.
16:02As the pendulum is pushed even more, something quite startling occurs.
16:07Its behaviour is now truly chaotic, displaying no apparent pattern.
16:11And yet, still subject to the laws of motion, it has started to follow what has come to be known, somewhat mystically, as a strange attractor.
16:21Strange attractors are the hidden patterns of order in disorder.
16:26The structure within the chaos.
16:28This is the strange attractor that lies behind Edward Lorenzi's early weather models.
16:33It's a computer representation of the rules which generated his apparently random results.
16:42When I first drew up the attractor, this was before the term attractor had been introduced.
16:47And I think in meteorology you could identify attractor with climate.
16:52It's a set of states which do occur, as opposed to those things that don't happen.
16:57Mapped out in an imaginary mathematical space deep inside the computer,
17:04those weather states that do occur trace out this beautiful and appropriately butterfly-like shape.
17:10Each point on the light blue lines represents a single condition of the world's weather.
17:16And as we follow the little red cursor, we are seeing the weather unfold in time in the computer's mind.
17:23There are an infinite number of possible pathways which the weather can follow on this, the Lorenz attractor.
17:29And yet, paradoxically, those pathways never cross, even as we swing from one lobe of the structure to the other.
17:36Where, then, is the chaos in such an apparently well-organized pattern?
17:41The chaos derives from the attractor's ability to behave like a mathematical food mixer,
17:47jumbling up the constituent elements of even a simple straight line
17:51until each point on that line is travelling in different directions.
17:55The line is stretched and folded repeatedly, like so much dough.
18:00This is, again, the butterfly effect at work.
18:04But Ed Lorenz also indicated that there are states of the weather system that do not occur,
18:10that are not allowed, and that the points representing those improbable states,
18:14shown here in red, as the edges of the cube,
18:17are like snow in the Sahara, or heat waves at the South Pole.
18:21If they happen at all, they can't last, and will very soon be pulled in towards the attractor,
18:26sucked in until the system is once again its old, unpredictable self.
18:32In the new science of chaos, it seems the closer you look, the more you see.
18:51Even a dripping tap, regular as clockwork, indeed the basis of many early time-keeping instruments,
18:58turns out to have a chaotic trick or two up its piping.
19:02It is one of many dynamic systems showing a type of behaviour that has convinced many scientists
19:08that the rules of chaos are universal.
19:12When you gently increase the water flow, the drips start coming in groups,
19:16then longer groups, then even longer groups, until the cascade is truly chaotic.
19:21Amazingly, even here it seems, there's a mathematical rule at work,
19:25a rule that describes one of the pathways from order to chaos,
19:29a rule that has come to be known as period doubling.
19:33In the period doubling route, you start off with the system being in a periodic state,
19:38a simple periodic state, and as you change the parameter,
19:41this periodic solution changes its period, it doubles in length.
19:46Then it quadruples as you increase the parameter further, then it eight tuples, and so on.
19:51This sequence occurs very, very fast, and very, very quickly the system becomes chaotic.
19:57Let me demonstrate it by this electrical circuit,
20:00which is based on a simple radio valve.
20:03Well, the circuit's off at the moment,
20:05so the state of the system is attracted towards the most simple possible sort of attractor
20:09you can have, a simple point attractor here.
20:11When I turn it on, the circuit will start to oscillate,
20:15and you'll see this state get attracted towards a periodic attractor.
20:19Here's the periodic attractor.
20:21You saw how quickly the state got attracted towards it.
20:24Now I'm going to change one of the settings in the circuit,
20:28and we'll see the attractor evolve and go through a sequence of period doublings.
20:32There is period doubled.
20:34Let's change it a little bit more.
20:38There's a period four.
20:40You see that now there are one, two, three, four revolutions to go round.
20:45Now let's change the parameter very, very slightly.
20:48And already we've got a strange attractor.
20:52In 1975, a young physicist was working on the problem of how frequently this doubling occurred
20:58in two totally different and unrelated systems.
21:01Although the now familiar dark regions of order in the unfolding chaotic cascade
21:06were no longer much of a surprise,
21:08when he compared the rate at which each different system was doubling,
21:11it was clear that something very odd was going on.
21:15I immediately realized it was the same convergence rate,
21:19and in fact those numbers were the same,
21:22which was a completely extraordinary result.
21:25Because what it meant was that there were two completely different problems
21:29that I had considered, and yet a number came out
21:32that in no way depended on either of the problems.
21:35And then of course you have to start wondering to yourself,
21:37what then does it depend upon?
21:39So I set about trying to figure out what it depended upon,
21:42which produced the theory of period doubling,
21:45and as its hallmark has the problem of the property of universality,
21:49which is exactly to say that the measurable things
21:53in any one of these very different systems is always precisely the same.
21:57And this constant about four and two thirds is one property that's the same
22:01for all of these different systems.
22:04To apply the lessons of chaos to complex systems,
22:09mathematicians draw pictures of the behavior.
22:12As we watch this circle distort into ever-new curious shapes,
22:16it's hard to believe that they're all, even the original circle,
22:19computer representations of that innocuous equation,
22:22x squared plus c,
22:24the equation that gave us video feedback.
22:27Simple rules, it seems, can produce fiendishly complicated results.
22:32There is complexity here that was implied but never realized by Newton,
22:37glimpsed but never resolved by Poincaré.
22:41But this is only part of a much greater object.
22:45The icon of chaos, its mascot perhaps,
22:48is this strange mathematical creature.
22:51It has been derived from the simplest of equations,
22:54and yet it has turned out to be the most complicated creation
22:57in the mathematical universe.
22:59Using the computer as our microscope to travel into the valleys and peaks
23:03of this extraordinary object known as the Mandelbrot set,
23:06we see literally infinite complexity, however deeply we probe.
23:11And that turns out to be one of the hallmarks of both chaos and nature.
23:16This unique and paradoxical mathematical marvel is a single image,
23:37a single image comprising an infinite universe of numbers.
23:41every shimmering speck of colour in each spiral and tail
23:45is itself a coded key to other unseen mathematical creatures,
23:49other new galaxies of colour and form.
23:52This is the Mandelbrot set.
23:55It lives on the frontier of a new geometry, fractal geometry.
24:07The basic elements of the language of fractal geometry
24:11were known to mathematicians about 50 years ago.
24:15But at that time they were considered to be mathematical monsters,
24:19objects only useful to discuss the limit of certain theories,
24:25but nothing basic in nature.
24:27And I think that's the true merit of Mandelbrot
24:30to have pointed out to us and to convince us
24:34that indeed these monsters are the right things to describe nature.
24:39The fractal is a geometric shape.
24:41A geometric shape having a very special property
24:45that as you look closer and closer and closer to it,
24:48you see essentially the same thing.
24:50Now let me contrast that with the straight line.
24:53As you come closer and closer and closer,
24:55you see always the same thing.
24:57It is true.
24:58Let me contrast with a circle.
25:00From a distance, it's a circle.
25:02As you come closer and closer and closer,
25:04it becomes a straight line.
25:05It changes.
25:06It changes.
25:07And the fractal is something like coastline of the country.
25:10As you come closer and closer,
25:11you see more and more of the irregularity of the coastline,
25:15but in a kind of general way,
25:18the coastline looks very much the same,
25:20whether you look at it from a very big distance
25:22or from a very small distance.
25:24So you see that two kinds of shapes
25:26have this property of invariance.
25:28The straight line, which we all know everything about,
25:32and some other shapes which are like coastlines.
25:35Well, the fractals are the shapes which have this property
25:39and are not straight lines.
25:51This computer-generated coastline is a fractal.
25:54It was created, like the Mandelbrot set,
25:56by giving a computer a set of formulae and codes
25:59that encapsulate the shape of a coastline.
26:01Remarkably, however much it is magnified,
26:05however closely you look,
26:07it always contains an identical degree of crinkliness.
26:11Many natural objects are fractal in this way.
26:17Every snowflake is different,
26:19although amazingly they all obey the same snowflake rules,
26:22variations on a six-sided theme.
26:25One of the snowflake rules is branching, a fractal process.
26:31On smaller and smaller scales,
26:34this computer crystal repeats nature's laws.
26:43Variations in those branching rules
26:45differentiate crystals from plants.
26:47As each grows, minute differences in form are magnified,
26:54the butterfly effect at work again.
26:58The slow geological crawl of mountain growth is recreated in seconds,
27:07conjured by the computer magician from fractal rules,
27:11reminding us of the dynamic nature of these events,
27:14however slow.
27:19In fact, we can now create a whole fractal world
27:22with fractal forests clothing fractal slopes in a fractal valley.
27:27Fractal rain from fractal clouds bathes a fractal globe
27:32resting in a fractal universe,
27:35and all of it in the mind of a computer.
27:44And yet, the most complex fractal of them all,
27:47the Mandelbrot set, obeys one of the simplest rules.
27:52We saw this set for the first time in the middle of the night,
27:54we being my assistant, myself,
27:56and frankly, at the beginning,
27:58we thought that we are dreaming
27:59or that something had happened,
28:01the mistake had happened,
28:03because never did we expect to find this wealth of structure
28:06as a result of formula,
28:08which was of a ridiculous simplicity.
28:13Hidden within the Mandelbrot set
28:14is all the wealth of variety that can be extracted from the equation x squared plus c.
28:20The rules of chaos govern it on an infinite scale.
28:23You could magnify it forever and never run out of detail.
28:26It's incredibly beautiful.
28:30If you look at it, if you do computer experiments with it,
28:36if you compute blow-ups of it,
28:38you always find incredibly beautiful structures
28:42which you have not expected or even seen before.
28:45So, in other words, one might say that the Mandelbrot set
28:49is the most complicated
28:51and maybe also the most beautiful mathematical object we have ever seen.
28:59All natural shapes are dynamic,
29:01distinct from the fixed and static geometry of man,
29:04growing and evolving over time.
29:06The unfolding patterns of nature are dictated by the ephemeral presence of chaos,
29:13of strange attractors,
29:15betrayed by the textures of bark and the branches of trees.
29:20Air currents twist water vapor into the fractal shapes of clouds.
29:25But it's one thing to generate images that resemble real objects,
29:29quite another to divine the rules that made them that way in the first place.
29:33This real fern branches repeatedly,
29:36but can we identify the rule that makes it grow in this way?
29:39This fern seems to be a perfect copy of the real thing,
29:42created by a set of rules in a computer.
29:44Again, the rules are simple and repetitive.
29:47They involve a sort of computer gymnastics called affine transformations.
29:52With them, apparently lifelike and yet wholly artificial images may be constructed.
29:58What then is an affine transformation?
30:01If I take this picture here,
30:04and I then bend it.
30:06Oh, no, I know a good way to tell you.
30:10If you're watching the TV,
30:11and you move to one side and look at it from here instead,
30:14look at the TV set from a different angle,
30:17you actually see an affine transformation of the image on the TV screen.
30:21A lot of the graphics that are used nowadays on TVs do kind of panning in pictures,
30:28squeezing out a page.
30:30They're various, very simple geometrical motions of images.
30:38One was the example of a reflection, that's an affine transformation.
30:42A rotation is an affine transformation.
30:44And then rather queerer things are them.
30:47If I take a picture drawn on graph paper,
30:50and then I skew one of the axes,
30:53as everything suddenly becomes deformed,
30:55I just skew one axis,
30:57that's actually carried out an affine transformation.
30:59The extraordinary fact is that if you keep only the transformations,
31:06and you throw away the image,
31:08you forget it,
31:09your mind is wiped clean,
31:11then you drive those transformations with a random number generator,
31:18you drive them.
31:20It turns out that always,
31:23you get back the same original image that you started from.
31:28Which is precisely how Dr. Barnsley created this computer image of a maple leaf,
31:33simply by following the maple leaf rules.
31:36But following even the simplest rules can produce an image of bewildering complexity.
31:41To play this game, the chaos game, choose three points and devise some rules.
31:47Rule one, take a starting point anywhere on the screen.
31:51Rule two, randomly choose a number from one, two or three.
31:56Rule three, move halfway from the starting point towards the number chosen in rule two.
32:02So select a starting point and run the rules.
32:06Take the new point and run the rules again,
32:09and again, and again, and again, and so on.
32:15Surprisingly, what emerges is not a filled-in triangle, but a fractal triangle.
32:21It's fractal because as we zoom into it, copies of the original are successively revealed,
32:28receding literally to infinity.
32:30In the jargon of chaos, this curious triangle is the attractor for those simple rules.
32:36It's a new mathematics which sheds light on the old and intransigent problems of physics,
32:42problems such as turbulence.
32:51In a slowly flowing stream, the eddies and tiny whirlpools die out almost as soon as they form.
32:57But as the stream quickens, the eddies live longer,
33:01until a moment comes when the turbulence grows,
33:03building upon itself to produce a flow that swirls and tumbles wildly.
33:08Disturbingly, turbulence seemed to rear its head wherever scientists looked,
33:26although little was known of how it occurred.
33:28For decades, the theories proposed by the Russian physicist Lev Landau
33:33seemed the closest to providing the answer.
33:38Landau had the idea that turbulence would occur through a sequence of transitions.
33:44Each transition would bring a new frequency into the motion,
33:48so the flow would become more complicated.
33:51And after many of these transitions, then the flow would appear very complicated.
33:56But fundamentally, it would be no different than it would be after one transition.
34:00It would still be described by a few frequencies.
34:04We had in mind testing this idea by looking in detail at the transitions
34:09to see each new frequency appear.
34:13It was hoped that Landau's transitions could be made to show up as patterns forming
34:18and dissipating in a thin layer of liquid trapped in the gap
34:21between two carefully controlled rotating cylinders.
34:25To show the patterns more clearly, tiny flecks of aluminium were added to the liquid.
34:31The question was, would the transition to turbulence be ordered,
34:36appearing at certain clear frequencies, as Landau suggested, or not?
34:40We saw that there was indeed a transition where a frequency came into the motion,
34:53some waves, and then another transition where more waves came into the motion.
35:02But then the next thing that happened was, as we increased the speed of rotation of the cylinder,
35:12the next transition was one in which the flow became irregular, chaotic.
35:17We were very excited by this because Landau was one of the great men of 20th century physics.
35:30He has enormous intuition into physical phenomena, and he was rarely wrong.
35:36So we recognized that if we saw something that was different from what Landau had expected,
35:44that this was something that would be exciting.
35:48And what we can show is that the flow really does have this property when it becomes irregular.
35:54That is, that the flow beyond this point of transition becomes inherently unpredictable.
36:00You can predict the behavior for the short period of time, but the long-term behavior is inherently unpredictable.
36:07No matter how accurately you measure the velocity or whatever the parameters are, you cannot predict the long-term behavior.
36:14What Harry Swinney's group have seen and measured are the milestones that occur on the unpredictable road to chaos.
36:23For many modern technologies, understanding how and when those milestones will occur is crucial.
36:30Whether it's ensuring that the flow of air over an aircraft's wing remains smooth,
36:36or that turbulent mixing with the atmosphere will occur for certain industrial emissions,
36:40a knowledge of the onset of chaos is vital.
36:45Surprisingly, it's now clear that we need look no further than our own bodies to find turbulence and chaos.
36:51Not only can turbulent blood flow damage blood vessels, but it can even affect the electrical activity of the heart itself.
37:01We have often seen the rhythmic electrical pattern associated with the normal ECG.
37:12But when the pattern changes, when the electrical rhythm fails, the heart fibrillates.
37:19This quivering, chaotic activity is more familiar to us as a heart attack.
37:25However, research in the USA on the ECG patterns of animals indicates that just before the heart begins to fibrillate,
37:33the normal pattern changes, displaying alternating high and low beats.
37:38When the research team at MIT first saw these new patterns in 1983, they called them electrical alternans.
37:47About that time, Mitchell Feigenbaum came and gave a lecture to the MIT Physics Department on period doubling.
37:55And since the pattern that we were noticing had a period which was twice the period of the normal ECG,
38:01which is just the normal intermediate interval, it occurred to us that perhaps this was a period doubling,
38:08which may be the first step in the progress to an apparently chaotic behavior.
38:14The immediate question was whether the period doubling that we had observed was in fact the first in a cascade of period doublings.
38:24What we did was to take 25 studies and to record on those patients the surface electrocardiogram
38:34and to use our computer algorithms to determine whether electrical alternans was present in those patients to a significant extent.
38:43We found that in about 90% of the individuals that he was able to induce serious arrhythmias in, indeed, electrical alternans was present.
38:54We would hope that we would be able to detect electrical instability on the order of six months or a year before an individual might sustain ventricular fibrillation.
39:06But this is really projecting far into the future. As I say, this research is an early stage and we're in the process of testing some of these hypotheses.
39:14And, you know, the technique is not in general clinical use at this time.
39:21No one knows how many heart attacks were caused in October 1929 when Wall Street crashed.
39:28It's now believed to have been the manifestation of chaos in a social system, triggering a cascade of calamity and misery.
39:36When it happened again in October 1987, economists turned to the new mathematics of chaos for some answers.
39:51There are very strong elements of chaotic phenomena involved in economic data.
39:57Consequently, that tells me that the forecastability or the predictability of economic events is strictly limited.
40:04Now, the question really is, what is the degree of limitation?
40:07How many time periods ahead can you really forecast?
40:11When does your predictability dial?
40:13Consider the following simple scenario.
40:19A group of people all have exactly the same preferences.
40:22They have the same wishes and desires.
40:24They all go to the same cinema to see the same show and want to sit in the same seat.
40:29And immediately, having all the same preferences and desires and the same information, try immediately to shift to the same alternative seat.
40:37What do you have? Chaos.
40:39You cannot do it.
40:41Similarly, for example, in the stock market, to be a little more realistic.
40:43If everyone is truly convinced, at the same time, that the price of a given stock is going to go up, then everybody wants to buy, no one wants to sell.
40:54You have a very unstable situation.
40:57So, here we have an irony by increasing the quality and speed of delivery of information of creating a situation which is potentially very unstable, suffering potentially from what we call a catastrophe.
41:10What happened October 19th was the culmination of a long period of a dynamical process that had, in its inherent structure, this very collapse.
41:23That is, a dynamical process was started, which inevitably was going to sort of hit a peak and then fall off, recover, and then sort of follow on a new path.
41:34That's what happened.
41:35And it is foolish, I think, to go looking for what I call a dot and a scapegoat.
41:39They aren't there.
41:40Or, alternatively, you could just think of all sorts of things that might possibly have precipitated it, only to discover that if you have this problem happy to gain, you'll have to look for a different set of alternative dot and scapegoats.
41:55There is a sort of ongoing joke in many academic circles amongst economists that, while one would hope that in ignorance, governments would at least half the time get their actions right,
42:08the irony seems to be that very seldom do they get their actions right.
42:12They take actions to alleviate a situation.
42:15You re-evaluate it several years later and discover the situation far being alleviated or improved is being made worse.
42:21Now, I'm not saying that the study of dynamics will eliminate that entirely.
42:27There are many other reasons leading to these problems having to do with the nature of the political process.
42:33But I think, in large part, a good deal of why we take inappropriate action is because we are acting in a linear fashion.
42:42But we are, in fact, in a non-linear world.
42:46We also live in a non-linear universe, a universe where even the orbit of a planet can be unpredictable.
42:53Imagine what it would be like to live on this planet orbiting not one, but two suns.
43:00With no predictable orbit, there would be no predictable day lengths and no predictable seasons.
43:05In fact, no constant cycles at all.
43:09It is difficult to see how life could evolve on such a planet.
43:15We are lucky that our own star system exists in a stable region of space,
43:20one of many islands of stability lying within chaos.
43:23Nevertheless, using space probes like Voyager, we are now discovering that despite Newton's laws,
43:33some planetary bodies bear the stamp of chaos.
43:38One of them is a small moon of Saturn called Hyperion.
43:42Hyperion mixes chaos and regularity in one object.
43:46What's happening is Hyperion is going around Saturn and the orbit around Saturn is just regular elliptical orbit, very predictable.
43:55If you want to know where it will be in orbit 100,000 years from now, then people can tell you and they'll be right.
44:00But if you want to know what direction it's pointing, that's much more difficult.
44:05It's like a potato, it's a very irregularly shaped thing and it tumbles chaotically and randomly, or apparently randomly, as it goes around the orbit.
44:12So it goes around regularly but tumbles chaotically.
44:16Paradoxically, Newton's laws allow not only for the existence of chaos within order, as shown by Hyperion, but also for the existence of order within chaos.
44:28The red spot of Jupiter is the most prominent atmospheric feature on this turbulent planet.
44:34It's an unlikely island of order within chaos.
44:36But then, as Harry Swinney points out, Jupiter is a most unusual place.
44:43There are horizontal bands which are jets of fluid, like the jet stream.
44:48If you look in detail, as was done in the Voyager photographs, the Voyager measurements, at the flow in these bands, it's highly turbulent.
44:56So the question is, if you look at Jupiter, you see large spots, vortices, that had been observed for a long time.
45:05In fact, the original observations go all the way back to a man named Robert Hooke in 1664, in which he observed a large coherent spot on Jupiter.
45:14So the question is, if you have such a spot, it's in the midst of a turbulent flow, why doesn't the turbulence just tear up this spot?
45:21How does it persist for centuries?
45:24And that's the question we set out to address.
45:26So we constructed an experiment, a laboratory experiment, a tank which is about one meter in diameter, which we rotate very fast to simulate the rapid rotation of Jupiter.
45:42We have the conditions which mimic, to a reasonable degree, the conditions of the atmosphere of Jupiter.
45:58And we observe the spontaneous formation out of a very turbulent flow of a single coherent spot, like the red spot on Jupiter.
46:06If there are two spots present, we can see that in a short period of time, they will approach one another, they are attracted to one another, and then they will merge, making a single spot.
46:29Like Jupiter's red spot, there are also some stable features in the Earth's atmosphere, yet mostly chaos seems to rule.
46:41Is there then no hope for accurate weather forecasting?
46:45With advancing computer technology, it appears there might be a way now of determining in advance whether the atmosphere is going to be predictable or chaotic over the period of interest,
46:56and therefore whether the weather forecast is going to be reliable or not.
47:02And that is, instead of running a single weather forecast from a single set of initial data, we would run an ensemble, a set of forecasts, in parallel from similar but not quite identical sets of initial data.
47:22If Tim Palmer's weather maps develop in similar ways, then he can conclude that that day's weather is in a predictable state, and that his forecast will be reliable.
47:32But if the weather patterns diverge, then the forecast will inevitably be unreliable.
47:37Here are two nearly identical weather systems, each showing a blue low pressure centre and a red high pressure.
47:46Once the computer begins to run its model forward, we can see that after ten days, in both cases, the pattern of isobar lines have stayed in step, resulting in high pressure over southern Europe and low pressure over Scandinavia.
48:00And in both cases, the weather of the UK would be very similar.
48:07In this second run, again, the initial pictures are similar, but this time, as the forecast is run forward, the right-hand map develops a strong low over the North Sea, while on the left, this low develops much further north.
48:18The final frame shows widely different pictures, suggesting that the initial weather state was unstable.
48:25The significance of Tim Palmer's work is that through chaos, he can now predict the limits of his predictions.
48:32Increasingly, chaos is teaching us what can, and perhaps more importantly, what cannot, be foretold.
48:38One of the main lessons that it teaches us is that we should be far more concerned and far more sceptical about the sort of long-term predictions and promises and forecasts
48:47which are offered to us about complex systems, such as the environment and the economy.
48:52The environment is essentially the weather system, and that is driven by the rotation of the Earth and the differential heating between the poles and the equator.
49:01And we are gradually changing that heating by the increasing pollution and the depletion of the ozone layer.
49:08Now, one lesson we learn from dynamics is that as you change the underlying conditions of a dynamical system, in this case the heating,
49:17then you can cause the attractor of the dynamical system, in this case the climate, to suddenly disappear.
49:25And one gets a sudden jump from one attractor to another, in this case from one climate to another.
49:30And so you see how with a slow and gradual change in the underlying conditions, you can cause a sudden and abrupt jump in the state of the system, in the climate.
49:39And that in this case would be an environmental catastrophe.
49:43The consequences of such a sudden catastrophe are almost unimaginable.
49:52A prospect is all the more chilling, because if it does happen, it will be the unpredictable result of an apparently insignificant action we will have taken.
50:00This is the lesson of chaos.
50:05But chaos can also provide an insight into how to detect and avoid the catastrophic, and how to measure and explain the commonplace.
50:12It has already changed the way scientists view the world, and could one day change the way we choose to run it.
50:19Central to this new perspective is geometry, and the gateway to geometry is the computer.
50:26Through it we may enter into the kingdom of chaos, a realm populated with mathematical monsters.
50:32A realm in which order and disorder are infinitely and inextricably interwoven.
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