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The butter fly effect is the idea that the tiny causes,
like a flap 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 61different movies,
TV episodes, and short films with ‘butterfly effect’ in the title
not to mention prominent references in movies like Jurassic Park,
更不用说像《侏罗纪公园》是歌里 书里 还有表情包
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
比如行星的运动 或是下落的物体 或是钟摆
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
比如温度 压力 湿度等等
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
问题是 和钟摆不同 这个系统是混沌的
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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Part of this video is sponsored by LastPass.