Tonight on NOVA, once we thought the universe ticked off its motions with stately regularity.
However, much of the universe is like this.
Up until recently, science had no explanation for these phenomena.
The atmosphere, the oceans, much of outer space, it basically is turbulent fluid.
So one could take the point of view that you don't understand anything about the world until you understand turbulence.
The world around us is being redefined by the strange new science of chaos.
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I'm going to show you how to play the chaos game.
It's very simple to play, but it has an extraordinary outcome.
I begin with a big sheet of paper with three large points marked on it, labeled 1 and 2, 3 and 4, 5 and 6.
Now what I do is I choose a point at random on the sheet of paper.
I'll shut my eyes to do that, and now I'll just wiggle the pen around and dab a point.
There it is. That's my starting point.
So now what I do is I roll the die, and it comes up with a 4.
So what I do is I go halfway from my first point to the big point labeled 3 and 4.
I'll mark that new point. That's my new point.
I roll the die. This time it comes up with a 5.
So I go halfway from my new point to the big dot marked 5 and 6, and I mark that point on the piece of paper.
Now, of course, if you do this, you should use a ruler and do things accurately.
You need to play it for about 24 hours or longer, and you need to play very fast.
And now I roll the die.
And this time it comes up with a 3, so I go halfway from my new point to the big dot marked 3, and that'll be about here.
Now, of course, if you keep on doing this for a long time, what would you expect?
You'd expect that you'd get an enormous mess of dots.
But that's not what happens. Something very extraordinary and different happens.
Speed up the game on a computer, and this emerges.
The Chaos Game has produced a complex and ordered geometrical structure.
The revolutionary idea that a chaotic process can give rise to order is destroying old definitions of chaos we've clung to for centuries.
The word chaos comes from the Greeks. They believe the goddess Gaia pulled the Earth out of the chaos of disorder and uncertainty.
Since then, science has associated chaos with randomness, the unpredictable, and built a framework to ignore or exclude it,
because it was an ordered picture of the universe we wanted.
The Earthrise, seen from the Moon with a man's feet firmly planted on it, remains a potent symbol to us of our ascendancy over an ordered and stable solar system.
Our architect was Isaac Newton. With his laws of motion, he gave us a calculator to determine the position and velocity of every heavenly body.
The wonderful order of the solar system was revealed to all in this beautiful mechanical replica, the Aurary.
This instrument is called the Grand Aurary.
It's a mechanical representation of the solar system, showing the motions of the planets from Mercury out to Saturn.
It epitomizes the ingenuity of clockmakers to show the motions of the planets so that you could predict where the planets would be at any one time,
and so it had a wide usage in showing how the universe was really a giant clockwork mechanism.
The planets revolve with stately synchronicity, unfolding the future out of the past.
Below the surface, a complex nest of clockwork gears dictate their interactions.
Newton had provided a mathematical calculus of the same clockwork precision to explain how the real solar system was as reassuringly propelled.
Newton's ideas were extended. If the universe ran like clockwork, then theoretically everything was knowable.
It's taken physics over a hundred years to destroy this cozy clockwork.
The development of Newtonian mechanics led to the idea that basically the entire universe could be predicted if only you could know enough about its present condition.
And in that concept, the only thing preventing us from making predictions about the future, accurate predictions about the future,
was simply that we don't know enough about the present situation.
But gradually in physics, this idea of complete predictability was eroded.
The first time it was eroded, I think, was in the middle of the 19th century with the development of the kinetic theory of gases, or the subject of what is called statistical mechanics.
It's why bingo is such a lottery.
Think of these balls as molecules in a gas.
Suppose we wanted to predict when ball 68 is going to pop up.
We'd have to calculate the outcome of the millions of random collisions between all the balls.
Since everything indirectly is going to affect the fate of 68, it's theoretically impossible.
We're defeated by the sheer force of numbers.
Complexity leads to uncertainty.
Nonetheless, we've assumed that within those limitations, Newton still stands.
That the more we can find out about complicated things, the more we can approach his idea of ultimate predictability.
The clock is symbolic of that Newtonian predictability because of the way it smoothly, regularly ticks off time.
Newton's world is a linear one, where you get out exactly what you put in, or a direct proportion of it.
So things change smoothly.
But chaos changes things drastically.
It makes even simple events impossible to predict because of a phenomenon called non-linearity.
Very few things are as linear as this space gun.
Fire it.
Now fire again using twice as much force.
The shell goes twice as fast.
That's linearity.
A camel in the desert.
Put a 250 pound weight on its back.
It will bend.
Now put another 250 pound weight on.
It will bend twice as much.
Now put a straw on it.
That's non-linearity.
A small force having an effect out of all proportion.
The straw that broke the camel's back.
Non-linearity explains the behavior of this balloon.
It's wildly erratic.
Unpredictable.
The air we pump in is lost continuously as it flies, interacting disproportionately with all the other variables.
Non-linearity makes even very simple systems impossible to predict.
This is a marvelous little toy that one of my students gave me a few days ago.
It's only a little bit more complicated than a pendulum.
A pendulum has one object that swings or oscillates.
Something like that.
But this one has another little gadget inside of it that can also spin.
And the two motions are coupled together.
Suppose we give it a little kick so it begins to oscillate.
There is a small magnet under here that gives it a periodic kick so it doesn't run down.
You see a very complicated motion developing.
In fact, if you keep your eye on the little space shuttle there, you'll see that it executes just an incredibly complicated trajectory,
which is in fact an example of chaos.
And if you were to try to put the equations, Newton's equations, for this little apparatus on the computer and predict its motion,
you would find that it would be impossible to do that for more than a few cycles of the oscillation.
Although the space shuttle is chaotic, the movement of the pendulum remains perfectly regular.
And there's nothing unusual about the vibrating platform which is driving this beam.
It's regular, too.
Chaos is not simply a game.
It's causing very worrying problems in engineering.
Well, here's another example of a chaotic system.
It's a flexible beam which is periodically excited at the base.
And these beautiful vibration patterns were known for over a century to engineers and scientists.
However, recently we've discovered some nonlinear modes in which the beam can oscillate from side to side.
And as we increase the periodic excitation of the base, the beam can go into rather violent and chaotic vibrations.
These vibrations have serious implications for real-world devices.
Flexible structures exist in aircraft wings, space structures, helicopter blades, bridges.
And in fact, it may even be that the destruction of the Tacoma Narrows Bridge many years ago may have been an example of chaotic vibrations.
In fact, all the supercomputers in the world cannot predict the motion of this beam or when it will become chaotic.
This makes it difficult for an engineer to know whether he's designed a structure safely or not.
This crude model is demonstrating how a satellite solar panel can go into fatal chaotic vibration.
The frightening thing is that no engineer could foresee this chaotic behavior.
We're still waiting for fusion energy because scientists can't control chaotic interactions in the hot ionized gas inside this reactor.
If only modern science had appreciated one crucial idea of the great French mathematician Henri Poincaré.
On celestial mechanics, he gave us a definition of chaos in 1892 while exposing Newton's feet of clay.
Newton had proved the stability of the solar system with an appallingly simple model, the gravitational interaction of two bodies.
His equations of motion worked.
The good news was hastily extrapolated to the universe at large.
Well, Newton believed in the stability of the model, but he still had his deepest questions, I guess, about the connection between the model and the real solar system.
So he believed that the solar system, which obviously had several more planets but also comets and other strange phenomenon,
that there, in order to keep it on a regular track, that it was necessary for God to push with his divine finger the reset button of the universe annually and keep things going the way they were supposed to go.
Mistrusting Newton's metaphysics, 19th century mathematicians attempted to extend Newton's equations to just three bodies.
Disaster. They could find no stable solution.
Poincaré bluntly reasoned that since Newton couldn't predict the outcome for three bodies, we couldn't prove the stability of the solar system either.
And if prediction was out, we would have to settle for description.
The juggler makes his three-body problem look easy.
When Poincaré tried to describe the real thing, he ended up with chaos, a vision of orbits that were as tangled as a ball of string.
It's so complicated, he said, that I cannot even attempt to draw it.
We can see what Poincaré was driving at by watching the unstable motion of this small planet around two large suns.
It's so erratic, any tiny error in our knowledge of where it started out will soon be magnified. Prediction becomes hopeless.
Nowadays we call it sensitivity to initial conditions. This is how Poincaré defined it.
It may happen that small differences in the initial conditions produce very great ones in the final phenomena.
A small error in the former will produce an enormous error in the latter.
Prediction becomes impossible.
This is a nice way to demonstrate the fundamental cause of chaos.
We have a little pendulum with a magnet on the end of a rod.
We release it, and it undergoes a very complicated motion because it is interacting with three fixed magnets glued to the base.
The fundamental cause of chaos is basically this. When the magnet approaches the fixed magnet, it can go either to the left or to the right.
Which way it goes depends very sensitively on exactly where it is when I release it.
Similarly, at any other point along its trajectory, which way it goes will depend on exactly where it happens to be at any particular moment.
That phenomenon, sensitive dependence on initial conditions, as we call it, is the fundamental cause of chaos.
It arises from the underlying nonlinearity of the equations describing the motion.
We have always wanted to understand and predict the weather, but sensitivity to initial conditions was to frustrate all our efforts.
In the 50s and 60s, the dream of improving weather prediction seemed about to come true.
Complex model building was possible for the first time, thanks to the modern computer.
Meteorologists knew that collating data from all over the Earth was not going to be easy,
but they also knew that the atmosphere behaved like a fluid in motion and was therefore governed by a version of Newton's laws.
The equations contained some nonlinearity, but they thought a linear approximation would be good enough to give them some useful parameters with which to chart the future from the past.
Well, the way that was most popular was what we might call a linear forecast.
You say that tomorrow's temperature at Boston is going to be six-tenths times today's temperature at Chicago plus one-third times yesterday's dew point at San Francisco or something like that.
You'd probably pick better predictors than this particular example, but this would be the general type of formula that they thought could be made to work as well as any other one, provided you put enough different terms in the equations.
There was a feeling that eventually linear formulas could be discovered that would do as good a job as anything else.
Those meteorologists had never appreciated Poincaré and his warning about nonlinearity.
They constructed their models.
An international network of weather stations was yielding accurate information on temperature, humidity, and barometric pressure.
The more data they had, they thought, the more accurate the forecast would get.
But progress in weather prediction proved disappointing.
Ed Lorenz decided to go the opposite route, toward simplicity.
Perhaps the weather appeared to be unpredictable not because of complexity, but because of something simple but profound in its dynamics.
So he chose a simple model.
In a cold beaker of water, there is no structure, but as you heat it, these convection rolls emerge.
Water heats at the bottom, becomes less dense, rises, cools, and becomes dense again.
And it's pushed around by the hotter water rising behind it.
In weather systems, those convection rolls are caused when air is heated close to the ground and rises before losing heat to the upper atmosphere.
Lorenz found they could be modeled by equations for three variables, x, y, and z, which describe the speed and temperature of the convection roll.
Lorenz programmed those equations into his computer, assigned initial values for the three variables, and started it running.
The computer allowed him to run months of his weather model in a matter of minutes.
Slowly, the plot rose and fell as the values fluctuated.
One day, some feature of his plot particularly interested him.
He decided to run the plot again by taking the values of x, y, and z at that point and reinserting them as the new initial conditions.
He left for a cup of coffee, allowing history to repeat itself.
And for a while it did.
Then the two traces began to diverge.
Things got worse and worse.
By the end of the run, what should have been a perfect copy was presenting two completely different versions of the weather.
I was rather dismayed at first when I came back an hour later after two months of simulated weather and found that it wasn't behaving the same way
because we'd had a lot of trouble with our little computer and I thought this was more computer trouble at first.
And then when I looked at it more closely, I realized that what was happening was that I hadn't put the same numbers in.
I put in rounded off versions of the numbers and these small differences had amplified, doubling every few days until they became so large that they drowned out the signal.
The computer was calculating values for x, y, and z to six places of decimals.
But the printout from which Lorenz took his new starting values only printed out to three places.
Sensitivity to initial conditions, chaos, had blown those tiny errors up out of all proportion.
First big question is does this little model really behave the same way as the atmosphere?
Or I should say does the atmosphere behave the same way as the model?
If it did, this would mean that long-range forecasting would simply not be possible.
This was a really exciting thing that came out of it as far as I was concerned.
Lorenz was quite correct.
We can never know the initial conditions accurately enough to prevent some tiny unnoticed error,
even the flapping wings of a butterfly amplifying itself and destroying our predictions.
That's why we never will dramatically improve our weather forecasting.
Lorenz actually calls it the butterfly effect.
Lorenz followed Poincaré.
He began drawing out the complex geometry of chaotic dynamics back in 1962.
But the significance of his discovery lay hidden from mainstream physics
in the arcane language of a journal of meteorology.
In 1977, Robert Shaw in the foreground and Peter Scott were colleagues in the physics department
at the University of California at Santa Cruz.
Oh, fixed point. OK. Should we speed it up?
Sure.
They were playing with an old-fashioned apparatus called an analog computer.
Why don't we try varying a parameter while it's going at this speed?
OK.
No silicon chips, no digitized memory banks.
Switch it off there for the moment. Change these capacitors.
Just an untidy mess of electrical wiring.
Smaller capacitors mean higher speed for solving the equations.
One day, a visiting professor handed them Ed Lorenz's weather equations.
They turned them directly into electrical circuitry.
Well, you know, when you first look at this, it looks like a tangle of spaghetti,
but it's actually three equations.
The X equation is on the left and is generally with the green wires,
and the Y equation is in the middle, generally with the red wires,
and the Z equation is on the right, generally with the yellow wires,
but as you see, there are some red wires that go back to the green wire part of the circuit,
and there are some yellow wires that go back to the red wire part of the circuit,
so the three equations are coupled together.
When it's actually running, the electrons are flowing through these wires.
It's made of simple operational amplifiers,
and so just resistors, capacitors, and amplifiers, little electronic amplifiers,
will then solve these equations,
and you see the voltage coming out here is then the solution of the equation.
But no one had prepared them for this, an extraordinary dynamical object
with a trajectory whizzing erratically to and fro.
They'd breathed life into Ed Lorenz's butterfly.
They'd had no idea that such simple equations could generate such complicated behavior.
The equations that govern this attractor are no more complicated intrinsically
than the equations that would govern, say, a clock.
So a clock is a perfectly predictable entity.
I mean, that's almost a definition of a clock,
and yet a similar degree of complexity can produce something that behaves in this fashion.
But why is it this shape? What is it actually doing?
And what is it telling us about the simple convection rolls with which Ed Lorenz modeled the weather?
This is another version of Ed Lorenz's convection roll.
Here, dots represent varying degrees of temperature.
Dynamicists conjure up an abstract three-dimensional space to describe it,
whose axes, x, y, and z, represent the three variables in Lorenz's equations.
The x variable describes the speed of the convection roll.
As it goes in the reverse direction, the roll reverses, too.
Now it's going counterclockwise.
y describes a temperature gradient across the convection roll from left to right,
and z describes a temperature inversion effect.
So any point in this three-dimensional space is a coordinate that combines values for all three variables
and describes what the whole system is doing.
Now let's see what happens when heat is applied to the resting fluid.
The coordinate shoots out to a position describing the new dynamics.
There, the roll settles down to a stable motion.
It's said to be attracted to this point.
If we increase the heat, we perturb it further,
but again it settles down to another stable rolling motion.
But if we apply still more heat, we reach a point where it never stabilizes.
It flips backwards and forwards quite unpredictably.
We've driven it into chaos.
And here's Lorenz's butterfly again.
Okay, and there's the atmosphere, turbulent atmosphere, bouncing up and down, unpredictably.
Chaos scientists call this a strange attractor.
Order and chaos combined.
The boundary of the attractor is fixed because it's a product of the three-weather equations.
Yet inside it, the behavior is totally unpredictable.
But that randomness is constrained.
The trajectory never strays outside the margin of the attractor.
These dynamics are demonstrating sensitivity to initial conditions.
It's very similar to what happens to a deck of cards in the hands of an expert croupier.
Think of the deck as an attractor.
Shuffling is the perfect randomizing device within it.
Two cards that were initially side by side could by now be separated anywhere in the pack.
This is what happens on a strange attractor.
This little cross of dots represents the original coordinates
and the rounded off coordinates in Ed Lorenz's weather experiment.
They are so close together, their dynamics can be described by a single trajectory,
at least initially.
But keep your eye on the traveling coordinate.
The single trajectory is being pulled apart.
Each of these coordinates emerges with its own trajectory.
Some go one way, some another.
One of them is now describing a completely different version of the weather.
This is why it is so intrinsically unpredictable.
Most of nature is composed of turbulent phenomena like the weather.
Air, gas and fluids are described by the same Newtonian equations.
Over a reasonable period of time, the flow rate in this river doesn't change very much.
Neither does the rock and the waterfall jump about.
Yet the water never takes the same path twice.
Turbulence like this has baffled scientists for centuries.
The entire universe is filled with fluids in turbulent motion.
The atmosphere, the oceans, much of outer space basically is turbulent fluid.
So one could take the point of view that you don't understand anything about the world until you understand turbulence.
For a while, this trail of cigarette smoke rises in a simple stream.
Suddenly, there's a transformation. It becomes turbulent, chaotic.
But what rules, if any, govern this sudden transition from order to disorder?
Gullub, with a partner, Harry Swinney, built this cylinder to drive fluid from simple flow to turbulence.
Inside is a smaller cylinder, which can rotate at different speeds,
putting an increasing shear force on water which is contained in the narrow ring around the outside.
Small translucent particles suspended in the water allow us to see structure when it emerges.
They start by rotating slowly.
As we increase the speed of the fluid, you will see the emergence of a pattern,
emergence of horizontal lines that mark the boundaries between donut-shaped water seas.
These donuts are stacked along the direction of the cylinder,
and we can see them now as they start to form at a very well-defined rotation rate of the cylinder.
Water is rolling in a spiral motion sideways and up and around the donuts.
As they increase the rotation rate to a well-defined level, these wavy lines develop.
A subsidiary fluid motion has superimposed itself on the first, moving at a different speed.
As they increase speed further, there is an abrupt change.
Okay, we've increased the speed here, and we see now for the first time the emergence of disorder,
emergence of chaos in the flow.
If you look on the small scale here, you can see the irregularities that mark the appearance of this disorder.
The rotation rate at which this disorder appears is very well-defined,
just as the previous transitions happened at very well-defined rotation rates.
The flow around the donuts is switching erratically to and fro, creating a chaotic,
turbulent interference pattern at the border between donuts.
When you drive natural systems into chaos, they all follow a very similar route.
Apply energy to a system, and it will respond by moving to a single dynamical position, or state.
But put a little more in, and the system gets more complex.
In a fluid, for example, you would see two patterns of motion.
Put more in, and at a clear point, there will be four.
In some systems, the dynamics will double again to eight.
And here's the transition to chaos.
Now the system is hopping randomly between many states.
This is the music of chaos.
A dripping tap may be a source of irritation to us, but it was a source of inspiration to Robert Shaw and Peter Scott.
In their hands, the familiar faucet turned into a landmark experiment in nonlinear dynamics.
The idea was put into their heads by a visiting West German scientist called Otto Roessler.
It prompted another string and ceiling wax experiment.
Simple, straightforward, periodic.
That's looking pretty stable.
Yeah, pretty quiet.
Okay.
Conventional physics has no explanation for the behavior of a water drop.
Period one.
Shaw and Scott, with a dynamical analysis, got far more out of it.
Why don't we just change it slightly here?
They watched the way each droplet filled, as if the water were being poured into a balloon.
The skin of the balloon, the elastic surface tension, stretches, but eventually the balloon drops off
and the surface tension snaps back towards the spout just before the next droplet forms out of it.
That snap back is the nonlinear element in this little continuous cycle of droplet formation.
As the drops fall, they interfere with a laser beam, so the time interval between them can be measured.
On the oscilloscope, the peak on the left is the preceding droplet.
The peak on the right is the next one.
So if the right-hand peak stays in the same position, it means the flow rate is periodic, regular.
Shaw's genius was to turn this flow rate into a simple two-dimensional map,
where the time interval between drops one and two is plotted against the time interval between drops two and three, and so on.
If it's regular, those points should all fall in the same spot, and they do.
Okay, let's go up to a little bit faster flow rate here, and we'll get it to a chaotic regime.
Hopefully in some chaotic regime.
So it's up, oops, there it is.
Regular dripping represents pretty boring dynamics.
But what is happening now that they've opened the tap a little more?
On the oscilloscope, you can see the right-hand peak is jumping around, all over the place.
The time interval between the drops seems to be varying quite randomly.
Or is there any underlying structure to it?
If it were completely random, this screen would simply fill up with a great mess of points.
But look, there's a structure emerging.
The points are constrained within a finite shape.
Scott and Shaw have captured one view of another strange attractor.
That strange attractor means even the apparently random behavior of a dripping tap has a hidden order lurking behind it.
The order of chaos.
Surprising amount of structure, and quite repeatable.
We could switch it off and switch it back on, and that same pattern would be there again.
Kind of amazing.
A little more math reveals the full attractor.
This simple bend forms when the water drops are regular.
But now two frequencies emerge.
And now four.
The drops are beginning to pitter patter.
They're becoming chaotic.
This attractor is named after their German muse, Otto Rossler.
This twist, or funnel, in the attractor means that the dynamics of the water droplets is becoming more complicated.
When the flow rate is increased, this regular cycle of stretch, drop off, snap back, and stretch again never has a chance to complete itself.
It goes out of phase, and since one droplet affects the formation of the next, the pattern becomes irregular, chaotic.
This analysis won't tell you why a system behaves as it does, but it shows you the whole range of how it behaves in a simple dynamical portrait, the attractor.
And since life is intrinsically dynamic, this power of description is proving very useful in medicine and biology.
This is a human heart filmed during major surgery.
It's beating normally.
But this wild, erratic behavior is similar to the onset of a dangerous pathology of the heart called fibrillation.
Hilda Elmond is recovering from heart surgery at Cedars-Sinai Hospital in Los Angeles.
She looks comfortable, but the upper chamber of the heart, the atrium, keeps jumping from normal rhythm to fibrillation.
It's a shock effect of drastic surgery she's just undergone.
Yeah, she's had an acute insult from surgery, from open heart surgery, and she's going to be in and out of atrial phase.
It's also hard to control their rate frequency.
Oh, look at that.
Now she's back in silence.
She's demonstrating just about every rhythm you can think of.
No self-respecting group of cardiologists these days seems replete without its resident nonlinear dynamicist.
Here, it's Alan Garfinkel.
Start that up again.
You know she's still in atrial phase.
No, she's not.
She's coming out.
Chaos theory may help them to understand why even healthy hearts can, apparently at random, start fibrillating.
A sudden onset of certain types of fibrillation can kill, and it's impossible using traditional techniques to predict when they're likely to happen.
That's atrial bigeminy.
As many as half of the cardiac deaths that occur every year are totally unexpected and sudden,
and the physician has no good predictors of what patients are at risk for such sudden death and what patients are not.
The heart is a mechanical pump where the contraction of the upper chambers, the atria, stimulates the ventricles.
It's driven by electricity.
The main pacemaker, the sinus node, sends an electrical wave across the atrium, making it contract.
That's received by the atrioventricular node in the center, which relays the signal down to the ventricles.
As long as this chain of electrical events is stable, the heart beats normally.
This is what it produces.
Each beat is a signature of one cycle of the mechanical activity of the heart.
It contains a surprising amount of detail.
First comes the P wave.
That's when the atrium is contracting.
Then there is the big QRS complex.
That's the power stroke of the heart when the ventricle contracts.
Finally comes the T wave, when the heart has to relax before it's ready to start the cycle again.
And it turns out that the T wave is the really interesting part.
In our work at Cedars-Sinai, we've been studying what happens to the heart as it becomes progressively destabilized, intoxicated, stressed.
And what we see is a very remarkable sequence of attractors, which are astonishingly like the Rissler sequence.
In the Rissler system, you recall, you see a very striking sequence of qualitative changes as that stress parameter is increased.
It begins with a simple oscillation.
It goes through period doubling to that banded structure.
It then goes through another bifurcation and develops a fold.
When it develops the fold, it becomes a little bit more chaotic, but it's still really quite regular within its chaotic action.
And then, as that stress parameter is increased even further, it develops a funnel.
And what happens in the funnel is that the state point was going around, gets periodically diverted into the funnel.
When it gets into the funnel, it gets whipped around a couple of times.
It comes back out into the oscillation, and it sort of doesn't know where it is anymore.
That has the effect of radically chaoticizing the behavior so that it's not periodic at all anymore.
When Garfinkel did a dynamical analysis of the heartbeat data, he came up with this attractor when the heart was only mildly stressed.
And this more chaotic attractor, very similar to the Roessler funnel, when the heart was on the brink of fibrillation.
Something as complex as the human heart behaves dynamically, just like a dripping tap.
If you think about what that trajectory in the phase plane is telling us,
actually telling us a very interesting thing about what causes the transition to chaos in this system,
you see two oscillations. You see a big loop, and you see that small loop with a funnel.
The big loop is the depolarization. It's the power stroke. It's the ventricular contraction.
And people rightly focus on that as the essence of what the heartbeat is about.
But what we see in this is that what causes the transition to chaos is the dynamic complexity that arises in the repolarization,
in the recovery, in the T wave. That's the little funnel.
And what's happening is that the recovery is becoming dynamically complex.
And so the state point going around gets trapped in a complex recovery and can't recover in time and in the right fashion.
That's what causes the real transition to chaos.
This heart is in atrial fibrillation. Highly irregular impulses are crashing like waves on the atrial ventricular node.
It's swamped. It never has time to recover before the next wave crashes in.
So the normal phase of the relay to the ventricles is destroyed, and they beat very irregularly,
just as the water drops float irregularly from the open tap.
Now that the heart is a nonlinear system and its dynamic behavior is very much similar to other nonlinear dynamic systems,
it is providing us a tool, a means of monitoring the rhythm from a different perspective.
It indicates that there is an underlying pattern to this irregularity, that even though it looks random, it is not random.
That there is an underlying mechanistic explanation for what is happening.
The cardiologists give their patients antiarrhythmic drugs to slow the electrical conduction rate of the heart and reduce the chance of fibrillation.
If these events are totally random, it makes it very difficult for the doctors to calibrate drug dosage with the heart's response.
But if heart fibrillation is chaotic, there should be a clear pathway to disorder, and a clear pathway back again.
We're also hoping that as we treat people with one of these antiarrhythmic drugs,
that the process that they underwent to achieve chaos gradually reverses, and eventually we end up with a normal sinus rhythm again.
If we route back to that normal sinus rhythm from chaos, we appear to be seeing repetitive cycles, lower dimensional attractors, if you will.
We're going from high dimension chaos to lower dimensional chaos, and hopefully eventually back to a regular sinus rhythm.
It's a new microscope. It's an aid in the same way the stethoscope is an aid to hearing,
in the same way the electrocardiogram is an aid to analyzing the pulse of the patient.
If it gives us a new way of looking at the patient, hopefully it will provide us new mechanisms and a new basis for predicting those people who are at high risk and those people who are not.
The heart is a rhythmic periodic organ, so chaos in the heart can be fatal.
But chaos is not always bad news.
Some biological systems thrive on it.
Here, the nervous system of a simple invertebrate animal, a medicinal leech, lies exposed to a microelectrode.
What we're doing here is eavesdropping on a single nerve cell inside of the leech,
and each of these spikes corresponds to a signal that's coming out of the nerve cell.
The nerve cell of a leech lives in a much more complex environment than a heart cell.
A heart cell sees a periodic or a fairly periodic kind of input.
In the case of a nerve cell, it's getting information from all sorts of different directions, and for that reason, its output is much, much more complex.
In the case of the heart cell, you can say, by very elegant computation, that it is obeying a fairly low-dimensional dynamical behavior.
In the case of the leech neuron, the behavior is much more complex, corresponds to a much higher dimension, but it is not truly random.
This is the strange attractor Rapp constructs from his nerve data.
It may look random, but it's a lot more constrained than a picture of total randomness.
Constrained randomness is the secret of healthy nerve action.
This is a patient with Parkinson's disease.
His valiant attempts at muscle coordination seem to us highly disordered or random.
Tremors in the neuromuscular system, be they Parkinsonian tremors or cerebellar tremors or the tremor that I'm inducing by fatigue lifting this object here,
may look random to the untutored eye, but in fact, if you look at it, they're in fact quite periodic.
We think they're produced by synchronization of motor units, a pathological synchronization of motor units,
which the nervous system under normal circumstances, I have to put this down, under normal circumstances,
the nervous system has mechanisms for desynchronizing motor units and preventing tremor.
And under pathological conditions like fatigue or Parkinsonianism or cerebellar lesions or various other diseases, this active desynchronization mechanism breaks down.
Nerves firing remind Garfinkel of a pan of hot bubbling cheese sauce.
This is our computer version of it.
In cheese sauce, each bubble suppresses the formation of other bubbles close to it,
preventing one disastrous volcano and spreading the bubbling out in a desynchronized manner.
Biological systems, at least in some respects, need to be chaotic.
They need to be bubbly. They need to stay loose.
And chaos can provide the mechanism whereby it does that.
In the muscle, we think that there is a similar mechanism.
Renshaw cells carry out exactly that same lateral inhibition that you see in the cheese sauce.
And the result of that in the healthy individual, in the non-tremor individual, is exactly that nice desynchronization of motor units.
They're firing, perhaps randomly looking, but not really.
They're firing in an actively desynchronized manner and thereby enabling you to avoid the spasmodic motion that you see in tremor.
Here are two brain waves. One looks ugly, irregular, looks very arrhythmic.
Here's another one that looks extremely regular, nice round wave followed by a sharp wave, round wave followed by a sharp wave.
Which one would you rather have?
The answer is that the upper one, the ugly, irregular one, is a normal adult EEG.
The bottom one is a spike wave complex of an epileptic seizure, which is a very dangerous and very pathological condition.
The moral of the story there is I think that order is not always good.
In brain waves at least, this kind of order is pathological, and a chaotic-looking EEG is a healthy one.
This experimental data looks quite normal.
An irregular, healthy EEG at the top, a regular heartbeat, and a smooth respiratory cycle.
Two of Garfinkel's colleagues have studied the effect of one drug, cocaine, on these rhythms.
In goes the cocaine.
Now the EEG resembles an epileptic fit.
The heart is fibrillating, and the breathing cycle is irregular.
One drug has caused pathological order in the brain and pathological chaos in the heart and lungs.
In a laboratory in Philadelphia, a subject rests quietly while his brain waves are being recorded.
It's called an electroencephalogram, EEG.
If a healthy brain is chaotic, it should be possible to find a strange attractor for mental processes,
though a skeptic might wonder what such a dynamical analysis could usefully tell you
about the endless chatter between billions of nerve cells that occurs when a human being is thinking.
One might consider it's analogous to hanging a single audio microphone over Beijing.
One could not possibly begin to understand conversations.
But in fact, there is some information to be learned.
If you were to record for a sufficiently long period of time, you would first of all discover
that there's about a 24-hour cycle to the sound that you're picking up.
And if you continue to record for even longer periods of time, there is a certain seasonal variation.
People are out in the evenings longer in the summer months.
And in fact, on a few very special occasions, such as a New Year's celebration,
there will be annually a very abrupt transition of behavior, which nonetheless reproduces itself.
So by hanging a microphone over Beijing, you are not going to learn Chinese,
but you could in fact learn something about the collective behavior of Chinese urban populations.
And this is exactly what one is doing electroencephalographically.
You take an electrical signal recorded from a subject at rest
and perform the kinds of mathematical analysis that one does for signals from fluid experiments
or electronic circuit experiments.
In fact, instead of getting a very complex random kind of pattern of behavior,
you get a very simple and visually very beautiful kind of a pattern,
which corresponds to a very low dimensional attractor.
To find out what happens when the subject is mentally more active,
they get him to count backwards to himself in sevens, a not so easy mathematical task.
They do a new dynamical analysis, and the attractor changes.
It becomes more complex, fatter, a higher dimensional chaos,
but it's still constrained by an attractor.
They hope there will be a series of attractors specific to different levels of mental activity.
Rapp thinks the brain embraces chaos because this unique combination of order and randomness
helps the problem-solving job it was designed to do.
One is compelled to ask the question,
what functional advantages can chaotic behavior actually provide?
One of the areas where it's fairly clear that advantages can arise is in the area of optimization.
Sounds complicated, but it's not.
Essentially, the notion of search, the search for the best solution.
What a certain amount of turbulence in that search can do is help you avoid becoming preoccupied
with a small-scale solution and possibly lose sight of a much better solution.
It's almost a forest-in-a-trees phenomenon.
What you want to be able to do in optimization is to scan a wide range of possible solutions
and to avoid locking on to a suboptimal solution early in your search.
One wants to have a certain amount of turbulence or randomness in that search.
However, this randomness has to be a controllable randomness
because eventually one wants to converge onto a solution.
You require something which seemingly is self-contradictory.
You require randomness because that's going to be quite helpful to you,
but it has to be controllable in some way.
That's exactly what a chaotic system does because it is extremely disordered,
profoundly turbulent, but it is a deterministic system.
It responds to controls in a deterministic way.
That means that in fact it will be subject to control by the central nervous system itself.
It's a beautiful biological device.
By learning to distinguish between utter randomness and the constrained randomness of chaos,
scientists like Paul Rapp are extending Newton's cause-and-effect determinism
into the strange new world where order and randomness combine.
Thanks to chaos, scientists are beginning to model the real world in all its nonlinear complexity.
The red spot of Jupiter has survived in the eye of a hurricane
since Galileo first noticed it in 1610.
It's puzzled scientists for centuries,
and until recently there were no convincing theories for it.
This is a computer simulation of the Jovian atmosphere,
and there's the red spot.
Scientist Philip Marcus programmed in Newton's equations,
added some terms for the Coriolis force, and some starting conditions.
The nonlinearity of chaos did the rest.
A stable, self-organized vortex rose spontaneously out of the turbulence.
Down in Texas, Harry Swinney has created chaos in a spinning bathtub.
His experimental expertise in fluid dynamics has recreated the turbulent conditions of the Jovian atmosphere.
The red dye will reveal the turbulent patterns to a spinning camera.
This is what it sees.
The red spot, now approaching 12 o'clock, is indeed a huge vortex.
Now it's approaching 3 o'clock.
Without the insights gained from chaos theory,
these experiments would simply never have happened,
and scientists would still be throwing out really interesting information,
because it appeared to be nothing more than random, noisy rubbish.
Chaos has enriched Newton's notions of cause-and-effect determinism,
and brought us closer to understanding the world as it really is.
Science is, after all, an enterprise of the solvable,
and it is simply the case that until these recent developments in mathematics
and computer technology became generally available,
these nonlinear problems were not, in fact, problems which we could solve.
That's one of the reasons why so many people are so very excited
about our emerging understanding of nonlinear dynamical systems and chaos.
What it has done is provide us with a language, a very rich and sophisticated language,
that we can bring to the analysis of complex, turbulent phenomena which surround us.
The domain of what we can investigate has been enormously expanded,
and this is extremely exciting to a scientist.
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