CHAOS 7- The Lorenz butterfly
When we look at the movement
of the atmosphere we soon
realize that it is
infinitely more complex
than that of the solar system.
The atmosphere is a fluid
that has, at each altitude
above each point of the Earth’s surface,
a speed, a density,
a pressure, a temperature etc..
All this varies over time.
It is of course unthinkable to know
all this infinite number of data.
It's almost as if we were in a space
with an infinite number of dimensions:
To understand something about it,
we must make approximations.
In 1963, Edward Lorenz
simplified, simplified,
and simplified the problem again.
He simplified it to such a degree
that there is no guarantee that
his equation still has
anything to do with reality.
His model of the atmosphere
was reduced to only
three numbers (x, y, z)
and the evolution of the atmosphere
to a tiny equation:
Each point (x, y, z) in space
symbolizes a state of
the atmosphere and the
evolution follows a vector field.
For example,
but this is only an example,
the first coordinate
could represent a temperature,
the second the wind speed
and the third the humidity.
Here, it is cold,
the wind blows and it rains ...
Here, the opposite is true ...
When we follow a trajectory of the field,
we are following the evolution of the weather ...
The forecaster just needs
to solve a differential equation!
This is what Lorenz saw
when he studied his model.
Does all of this have
anything to do with real weather?
Not clear at all!
It is what physicists often call
a toy model,
that is used to try and
understand the broad outlines
of a some complex behavior.
In fact, Lorenz had only
these sorts of graphs
to look at because
his computer, in 1963,
was quite primitive.
Let’s look at two atmospheres,
represented by the centers of these
two balls that are extremely close together,
so that they are almost identical,
and let’s observe what happens to them.
At first the two evolutions
remain indistinguishable.
But then they
split up significantly:
both atmospheres become
completely different.
This is chaos : the sensitive
dependence on initial conditions.
In 1972, Lorenz was going to present
his work at a prestigious conference,
but he was late in sending
the title of his lecture.
The organizer was in a hurry
to send the program to the participants,
and so he chose the title:
“Predictability : Does the flap
of a butterfly’s wings
in Brazil set off
a tornado in Texas?”
The butterfly effect was born!
Mathematical concepts are
not often well understood
by the general public
The image of a small,
frail butterfly having
an influence on the world is very poetic !
The idea was very successful
and was modified.
There is always a butterfly,
but sometimes it comes from Africa,
and sometimes from China.
..and it is responsible for disasters
in New York…or Chicago.
Interesting...
The idea found its way
into literature, music and movies.
The list of films
based upon it is endless.
A lot of emotions in Babel,
the 2006 film by
Alejandro González Iñárritu
The insignificant incident of
a gunshot in Morocco
will change the lives
of an American couple,
a Mexican nursemaid
and a Japanese girl.
An interplay of small causes
and large consequences.
..how chance
determines the fate of humans.
But unfortunately, only
one half of Lorenz’ message
made its way to the general public.
Can chaos theory
be limited to the statement
that it is impossible to
predict the future in practice?
How can a scientist resign himself
to such an admission of failure?
Lorenz’ message
is a lot richer.
Here are two trajectories
of the Lorenz system,
one blue and one yellow,
that do not necessarily
take off from close initial conditions .
Let’s erase, say, the first ten seconds,
of the movement and
let’s observe what follows….
What do we see?
That the trajectories are
indeed very different,
they seem a little crazy and
quite unpredictable.
But they accumulate
on the same butterfly-shaped object
that does not seem to
depend on the initial position.
They seem to be attracted to this butterfly.
That is why we speak of
the Lorenz attractor.
A strange attractor!
Now this is a positive scientific phenomenon,
not as famous as the butterfly effect.
Instead of observing just two trajectories,
let’s look at a lot more.
Look at all these balls, each one
of them representing a simplified atmosphere.
After a while,
they all accumulate
on the same butterfly.
A very nice object,
that one can admire endlessly.
These are real problems
for mathematicians
and scientists in general.
Instead of describing the future
starting fgrom a given initial condition,
which we know is futile,
we will try to describe the attractor.
What does it look like?
How do the internal dynamics work?
In the 1970s,
Birman Guckenheimer and Williams
proposed a simple model
for trying to understand
the Lorenz attractor.
Here is a strip of paper,
Let’s fold it, and cut it …and glue it together.
..and we have a special object in space!
Let go to the starting line.
Here we go,
we drive and little while later,
we're back on the starting line ...
But we find ourselves on another point
from which a new path starts.
The position on the starting line
is given by a number x
between 0 and 1.
At zero we are on the left,
at 0.5 in the middle and at 1 on the right.
If we start from the point x,
then the return point is 2x if x is smaller than one half,
and 2x-1 if it is larger than one half.
In other words,
when the car moves, the
points where it meets the starting line
are doubled at each turn,
except that we need to subtract 1
if the result is greater than 1.
It's a bit like the horseshoe :
the dynamics in continuous time
got replaced by dynamics
in discrete time, namely
the successive passages
on the starting line.
Look,
we start off from 1/3,
we arrive in 2/3,
then 4/3 but we must subtract 1,
that is to say 1/3:
We are back on the starting point
after two rounds.
So we have a periodic trajectory
of period 2.
Here is a trajectory of period 18.
A periodic trajectory
passes successively
through the left and right ears
following a certain sequence.
Well, it can be shown that
all sequences are possible.
And of course,
not all trajectories are periodic.
Whatever the infinite sequence
of left and right,
there is a trajectory
that follows this destiny!
What is the relationship between
the Lorenz model and that made of strips of paper?
Well, it was’nt until 2001
that the mathematician Tucker
showed that the paper strip model accurately
describes the movement on the Lorenz attractor.
For each trajectory
in the Lorenz attractor,
there is a trajectory
on the paper model
that behaves in exactly
the same way.
Can understanding the movement
of the atmosphere then
be reduced to continually
multiplying by 2
a number between 0 and 1,
and subtracting 1
if the result is greater than 1?
Of course not!
All this is too simplistic,
but it is an illustration of the phenomenon.
Simple things?
Mathematicians love them!
Why has the butterfly effect
become so popular?
Perhaps because it
gives us back our personal freedom.
The legacy of Newton’s cold determinism
could lead to a kind of fatalism.
The Lorenz butterfly claims,
small as we are, that
we can have an influence on the world!
Good news for us!