CHAOS 6- Smale in Copacabana
In the early 1960s,
the young American mathematician Steve Smale
was working on the beach of Copacabana
when he made a discovery ...
He discovered a horseshoe!
Not in the sand of course!
In fact, the horseshoe
is an abstract mathematical object!
It's another simple idea
that attempts to reduce chaos
to its most elementary expression.
We must first explain an old idea
dating back to Poincaré.
Here is a vector field in space.
A trajectory that starts on this disc
makes a full turn
and then revisits the disc
then another turn and so on.
For each point x of the disc,
we can observe the trajectory
that starts from that point,
and wait until it hits the disc again
in a new point T(x).
A few seconds later,
this new point is again transformed by T,
and we get a third point T(T(x)).
…and so on:
at each turn, the point is transformed by T.
Instead of studying the trajectory in space,
we study a sequence of points in the disc:
it's easier to draw!
We replace
the dynamics of the field in continuous time
by the dynamics of the transformation T
in discrete time: 0, 1, 2, 3, etc..
This is often much easier to understand.
Here the transformation T is a rotation by
one third of a turn.
The central trajectory is corresponds to
the fixed point of the rotation: the origin.
Other trajectories
are periodic also,
but it takes them three times longer
to return to the starting point:
the rotation is one third of a turn.
Here, the transformation T is a similarity.
The trajectories are no longer periodic:
On the contrary, they approach
the central trajectory while spiraling around it.
The central periodic trajectory is stable.
And here is an unstable trajectory:
The transformation T
crushes the kitten along its width
and stretches it along its height.
The center is a fixed point,
so its trajectory is periodic.
You see that the trajectories
located on the horizontal axis
approach the fixed point
while those on the vertical axis
move away from it.
Poor little cat!
So let’s go on to the horseshoe ...
Watch how this square
is deformed over time.
The transformation T combines
a scaling and a contraction,
but also a folding.
When the square arrives back
on the original disk,
it has taken the form of a horseshoe.
Now watch the behavior of the square
as we go back in time,
by the inverse transformation.
A scaling, a contraction, a folding,
but all in the other direction.
Let’s try to envision the future and the past of the square.
After one round, as we have seen,
it became a horseshoe.
After one turn the other way,
towards the past, it becomes a horseshoe,
turned by a quarter turn.
The two small vertical stripes you see,
at the intersection of
the square and the horseshoe
are also compressed,
scaled and folded.
The behavior in the past is the same.
And this whole structure is repeated an infinite number of times.
A complicated horseshoe!
How can we understand its dynamics?
Let’s call this area A.
and this one B .
Well, the amazing thing is
that we get almost the same results
as Hadamard’s geodesics and billiards :
For each finite sequence of A’s and B’s,
even when allowing repetitions,
for instance BABB,
there is a periodic point
that will follow this route exactly.
In the example, it is a point of B that jumps to A,
then jumps to B and then jumps to B again ,
and then jumps back to the starting point.
Here is the only point of A
that always remains in A:
it is a fixed point
This fixed point is in B.
Here is one that alternates between A and B.
But, as with the billiards, this
is not limited to finite sequences.
Give me an infinite sequence of your choice
BBAABBBAABABA etc.. etc..
Well, there is a point following
exactly this route in its future ...
When you think about it,
this is incredible:
every possible future
is true for at least one person…
(Oops, I meant for at least one point!)
Prediction is impossible
because everything is possible.
Free will rediscovered!
Anything is possible,
there’s a fine slogan for chaos!
Thank you Mr. Smale.
Smale realizes that
it is not always necessary
to know things exactly
in order to understand them.
A distorted picture is often enough
to recognize a face.
He shows that the horseshoe is stable.
Mind you,
this does not mean
that the trajectories are stable:
on the contrary, the trajectories have
a big sensitivity to initial conditions.
What we mean is structural stability.
Distort Mr. Smale’s picture, and you will still recognize him.
On the right we have the original horseshoe.
On the left another horseshoe,
slightly modified,
is being formed.
After all, when you stretch the square
into a rectangle
and then fold it in the form of a horseshoe,
you can do it in several ways.
This is somewhat analogous to a situation
where we would study two solar systems
where the planets have
slightly different masses
Two almost identical horseshoes.
We can do exactly
the same as we did
with the original horseshoe
and draw areas A 'and B' the same way.
And it can be shown in the same way
that the trajectories of this new horseshoe
are still described by sequences of A’s and B’s .
The modified horseshoe is
just as chaotic as the original one!
Chaos is really there,
indestructible.
In fact, Smale shows that
the movements on the two horseshoes
are somehow "identical".
Look,
the two trajectories, right and left
follow the same choreography,
they visit the same A, A 'and the same B, B'.
We can match a trajectory
of the second horseshoe
to any trajectory on the first and vice versa.
It is in this sense
that the structure is stable:
we can deform the horseshoe any way we want,
not only does it remain chaotic
but it keeps the same dynamics as before.
Individual trajectories are unstable
but the dynamics as a whole are stable.
Early in the twentieth century
Poincaré, with characteristic flair,
explains how it is possible to understand a dynamic
even if one does not know it very well.
«You ask me to predict
the phenomena that will occur.
If I had the misfortune to know
the laws of these phenomena,
I could only get there
by intractable calculations
and I would not be able to give you an answer;
but, as I am so lucky as to be ignorant of them,
I'll answer right away..
And what is most extraordinary,
is that my answer will be correct.»
The coexistence of chaos,
and hence the instability of individual trajectories,
and structural stability, a global property,
is absolutely remarkable.
I'm unstable,
but the world around me is stable!
Reassuring ;-)