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  • Part of this video is sponsored by LastPass.

  • More about last pass at the end of the show.

  • The butter fly effect is the idea that the tiny causes, like a flay of a butter fly's wings in Brazil,

  • can have huge effects, like setting off a tornado in Texas

  • Now that idea comes straight from the title of a scientific paper published nearly 50 years ago

  • and perhaps more than any other recent scientific concept, it has captured the public imagination

  • I mean on IMDB there is not one but 61

  • different movies, TV episodes, and short films with 'butterfly effect' in the title

  • not to mention prominent references in movies like Jurassic Park, or in songs, books, and memes.

  • Oh the memes

  • in pop culture the butterfly effect has come to mean

  • that even tiny, seemingly insignificant choices you make can have huge consequences later on in your life

  • and I think the reason people are so fascinated by the butterfly effect is because it gets at a fundamental question

  • Which is, how well can we predict the future?

  • Now the goal of this video is to answer that question by examining the science behind the butterfly effect

  • so if you go back to the late 1600s, after Isaac Newton had come up with his laws of motion and universal gravitation,

  • everything seemed predictable.

  • I mean we could explain the motions of all the planets and moons,

  • we could predict eclipses and the appearances of comets with pinpoint accuracy centuries in advance

  • French physicist Pierre-Simon Laplace summed it up in a famous thought experiment:

  • he imagined a super-intelligent being, now called Laplace's demon,

  • that knew everything about the current state of the universe:

  • the positions and momenta of all the particles and how they interact

  • if this intellect were vast enough to submit the data to analysis, he concluded,

  • then the future, just like the past, would be present before its eyes

  • This is total determinism: the view that the future is already fixed,

  • We just have to wait for it to manifest itself

  • I think if you've studied a bit of physics, this is the natural viewpoint to come away with

  • I mean sure there's Heisenberg's uncertainty principle from quantum mechanics,

  • but that's on the scale of atoms;

  • Pretty insignificant on the scale of people.

  • Virtually all the problems I studied were ones that could be solved analytically

  • like the motion of planets, or falling objects, or pendulums

  • and speaking of pendulums I want to look at a case of a simple pendulum here

  • to introduce an important representation of dynamical systems, which is phase space

  • so some people may be familiar with position-time or velocity-time graphs

  • but what if we wanted to make a 2d plot that represents every possible state of the pendulum?

  • Every possible thing it could do in one graph

  • well on the x-axis we can plot the angle of the pendulum,

  • and on the y-axis its velocity.

  • And this is what's called phase space.

  • If the pendulum has friction it will eventually slow down and stop

  • and this is shown in phase space by the inward spiral --

  • the pendulum swings slower and less far each time

  • and it doesn't really matter what the initial conditions are,

  • we know that the final state will be the pendulum at rest hanging straight down

  • and from the graph it looks like the system is attracted to the origin, that one fixed point

  • so this is called a fixed point attractor

  • now if the pendulum doesn't lose energy, well it swings back and forth the same way each time

  • and in phase space we get a loop

  • the pendulum is going fastest at the bottom but the swing is in opposite directions as it goes back and forth

  • the closed loop tells us the motion is periodic and predictable

  • anytime you see an image like this in phase space,

  • you know that this system regularly repeats

  • we can swing the pendulum with different amplitudes,

  • but the picture in phase space is very similar, just a different sized loop

  • now an important thing to note is that the curves never cross in phase space

  • and that's because each point uniquely identifies the complete state of the system

  • and that state has only one future

  • so once you've defined the initial state, the entire future is determined

  • now the pendulum can be well understood using Newtonian physics,

  • but Newton himself was aware of problems that did not submit to his equations so easily,

  • particularly the three-body problem.

  • so calculating the motion of the Earth around the Sun was simple enough with just those two bodies

  • but add in one more, say the moon,

  • and it became virtually impossible

  • Newton told his friend Haley that the theory of the motions of the moon made his head ache,

  • and kept him awake so often that he would think of it no more

  • the problem, as would become clear to Henri Poincaré two hundred years later,

  • was that there was no simple solution to the three-body problem

  • Poincaré had glimpsed what later became known as chaos.

  • Chaos really came into focus in the 1960s,

  • when meteorologist Ed Lorenz tried to make a basic computer simulation of the Earth's atmosphere

  • he had 12 equations and 12 variables, things like temperature, pressure, humidity and so on

  • and the computer would print out each time step as a row of 12 numbers

  • so you could watch how they evolved over time

  • now the breakthrough came when Lorenz wanted to redo a run

  • but as a shortcut he entered the numbers from halfway through a previous printout

  • and then he set the computer calculating

  • he went off to get some coffee, and when he came back and saw the results,

  • Lorenz was stunned.

  • The new run followed the old one for a short while but then it diverged

  • and pretty soon it was describing a totally different state of the atmosphere

  • I mean totally different weather

  • Lorenz's first thought, of course, was that the computer had broken

  • Maybe a vacuum tube had blown.

  • But none had.

  • The real reason for the difference came down to the fact that printer rounded to three decimal places

  • whereas the computer calculated with six

  • So when he entered those initial conditions,

  • the difference of less than one part in a thousand

  • created totally different weather just a short time into the future

  • now Lorenz tried simplifying his equations and then simplifying them some more,

  • down to just three equations and three variables

  • which represented a toy model of convection:

  • essentially a 2d slice of the atmosphere heated at the bottom and cooled at the top

  • but again, he got the same type of behavior:

  • if he changed the numbers just a tiny bit, results diverged dramatically.

  • Lorenz's system displayed what's become known as sensitive dependence on initial conditions,

  • which is the hallmark of chaos

  • now since Lorenz was working with three variables, we can plot the phase space of his system in three dimensions

  • We can pick any point as our initial state and watch how it evolves.

  • Does our point move toward a fixed attractor?

  • Or a repeating loop?

  • It doesn't seem to

  • In truth, our system will never revisit the same exact state again.

  • Here I actually started with three closely spaced initial states,

  • and they've been evolving together so far, but now they're starting to diverge

  • From being arbitrarily close together, they end up on totally different trajectories.

  • This is sensitive dependence on initial conditions in action.

  • Now I should point out that there is nothing random at all about this system of equations.

  • It's completely deterministic, just like the pendulum

  • so if you could input exactly the same initial conditions

  • you would get exactly the same result

  • the problem is, unlike the pendulum, this system is chaotic

  • so any difference in initial conditions, no matter how tiny,

  • will be amplified to a totally different final state

  • It seems like a paradox, but this system is both deterministic and unpredictable

  • because in practice, you could never know the initial conditions with perfect accuracy,

  • and I'm talking infinite decimal places.

  • But the result suggests why even today with huge supercomputers,

  • it's so hard to forecast the weather more than a week in advance

  • In fact, studies have shown that by the eighth day of a long-range forecast,

  • the prediction is less accurate than if you just took the historical average conditions for that day

  • and knowing about chaos, meteorologists no longer make just a single forecast

  • instead they make ensemble forecasts,

  • varying initial conditions and model parameters

  • to create a set of predictions.

  • Now far from being the exception to the rule, chaotic systems have been turning up everywhere.

  • The double pendulum, just two simple pendulums connected together, is chaotic

  • here two double pendulums have been released simultaneously

  • with almost the same initial conditions

  • but no matter how hard you try,

  • you could never release a double pendulum and make it behave the same way twice.

  • its motion will forever be unpredictable

  • you might think that chaos always requires a lot of energy or irregular motions,

  • but this system of five fidgets spinners with repelling magnets in each of their arms is chaotic too

  • At first glance the system seems to repeat regularly,

  • but if you watch more closely, you'll notice some strange motions

  • a spinner suddenly flips the other way

  • Even our solar system is not predictable

  • a study simulating our solar system for a hundred million years into the future

  • found its behavior as a whole to be chaotic

  • with a characteristic time of about four million years

  • that means within say 10 or 15 million years,

  • some planets or moons may have collided or been flung out of the solar system entirely.

  • The very system we think of as the model of order,

  • is unpredictable on even modest timescales

  • So how well can we predict the future?

  • Not very well at all at least when it comes to chaotic systems

  • The further into the future you try to predict the harder it becomes

  • and past a certain point, predictions are no better than guesses.

  • The same is true when looking into the past of chaotic systems and trying to identify initial causes

  • I think of it kind of like a fog that sets in the further we try to look into the future or into the past

  • Chaos puts fundamental limits on what we can know about the future of systems

  • and what we can say about their past

  • But there is a silver lining

  • Let's look again at the phase space of Lorenz's equations

  • If we start with a whole bunch of different initial conditions and watch them evolve,

  • initially the motion is messy.

  • But soon all the points have moved towards or onto an object

  • the object, coincidentally, looks a bit like a butterfly.

  • it is the attractor

  • For a large range of initial conditions, the system evolves into a state on this attractor

  • Now remember: all the paths traced out here never cross and they never connect to form a loop,

  • If they did then they would continue on that loop forever and the behavior would be periodic and predictable

  • so each path here is actually an infinite curve in a finite space.

  • But how is that possible?

  • Fractals. But that's a story for another video

  • this particular attractor is called the Lorenz attractor,

  • Probably the most famous example of a chaotic attractor

  • though many others have been found for other systems of equations

  • now if people have heard anything about the butterfly effect,

  • it's usually about how tiny causes make the future unpredictable

  • but the science behind the butterfly effect also reveals a deep and beautiful structure underlying the dynamics

  • One that can provide useful insights into the behavior of a system

  • So you can't predict how any individual state will evolve,

  • but you can say how a collection of states evolves

  • and, at least in the case of Lorenz's equations,

  • they take the shape of a butterfly

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  • and I know that is incredibly dangerous because if even one of those sites got hacked,

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  • Quite the butterfly effect.

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Part of this video is sponsored by LastPass.

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バタフライ効果を支える科学 (The Science Behind the Butterfly Effect)

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    林宜悉 に公開 2021 年 01 月 14 日
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