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Language: en
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of the show the butterfly effect is the
idea that tiny causes like a flap of a
butterfly'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 bass
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 eight
and kept him awake so often that he
would think of it no more the problem as
would become clear to Henri Poincare a
two hundred years later was that there
was no simple solution to the three-body
problem blank array 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 the
results diverged dramatically Lorenza
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 lorenza'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 equation
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 evolve and at least
in the case of lorenza's equations they
take the shape of a butterfly
[Music]
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