CHAOS 9-Chaos research today
There are many kinds of dynamics.
Some are complicated,
others are not.
Watch this vector field in space
that depends on a parameter a.
Let’s start by choosing a = 0.3
and let’s look at what happens
for a large number of initial conditions.
After a while,
they all fall on a periodic trajectory.
No chaos there.
There is an SRB measure of course as
everyone eventually
turns around on a closed curve.
Let’s change the parameter slowly.
Look, for a = 0.335
the periodic trajectory splits up...
Still no chaos, but a periodic trajectory
that is twice as long...
And then for a = 0.380,
the trajectory splits up again and is now fourfold.
Then things accelerate.
For a = 0.405
things become complicated and chaotic.
But what is surprising is
that if we continue
to increase the parameter a,
sometimes, without warning,
the chaotic dynamics
become simple and
falls back to a periodic orbit.
How can we understand these bifurcations?
What is the most common
behavior in nature?
Chaotic? Non-chaotic?
It is not clear at all.
For centuries we had
no idea that chaos could exist...
and today we see it everywhere!
For each value of a,
we can mark the trace of the attractor
on a plane and we draw it in red.
As the parameter varies we obtain
this drawing that looks like a piece of lace,
and this is called a bifurcation diagram.
Nice... but not easy to understand!
Here are some more dynamics.
Mathematicians seek to establish
results that are always valid,
but they often start
by studying examples.
They then hope that
what they see in a simple example
can be realized generally, for all cases.
Remember the idea of Lorenz,
Sinai-Ruelle-Bowen, SRB.
The proportion of time that a
trajectory spends in a ball
converges to a limit that
doesn’t depend on the initial condition.
Is it reasonable to expect
that this is ALWAYS true?
That would be too nice...
So unfortunately the answer is no,
as we will see on a small example
discovered by Bowen.
Watch this vector field in the plane.
There are three equilibrium positions.
A trajectory starts...
and after a while,
it approaches an equilibrium position
and then moves on to another one.
When it approaches an equilibrium,
it slows down and remains
in its vicinity for some time.
Then it restarts towards the other
equilibrium and remains there even longer.
Back to the first equilibrium
where there is an
even longer delay…etc…etc
So here is a little green disc.
Let’s observe the transit time in this disc.
Well, the trajectory stays there
for long periods so that the
proportion of the time is close to 100%.
Then we leave the disc for a very, very long time.
The proportion falls to a value close to zero,
then it goes back to nearly 100% and then falls back and so forth.
You see, there is no convergence.
There is no Sinai-Ruelle-Bowen measure!
So what can we do?
Abandon this idea?
Say that Lorenz was mistaken?
Well no!
It should be understood that the example
we have just seen is very special.
If we change the vector field just a little,
like this:
…there is no longer a problem with the new field.
The new trajectories
travel in the plane and eventually
approach a periodic trajectory,
so that statistically
everything behaves periodically
and now we do have a Sinai-Ruelle-Bowen measure.
So the question is whether
these measures show up,
not for ALL of dynamical systems,
But here again, we must
temper our enthusiasm...
Watch this vector field.
Here is a trajectory.
All is well, it accumulates on a butterfly,
as we have seen before...
If the initial condition moves slowly
on the blue axis,
everything continues as it should: the trajectory
continues to accumulate on the same attractor,
even if it takes a little while to get there.
And we could easily check that
the statistics haven’t changed.
But all of a sudden, without warning, surprise...
the trajectories accumulate elsewhere…
on another attractor
that has nothing to do with the first!
Space appears to be
split into two regions:
if one starts with a point in the first region,
the trajectory accumulates on the first attractor,
and if we start in the second region
we end up on the other...
In other words, we cannot say
that there is an SRB measure since
the long-term behavior depends on the initial condition.
There are two SRB measures!
For certain initial conditions,
statistics follow the first
and for others the second!
In the 1990’s,
the Brazilian mathematician Jacob Palis
formulated a whole set of problems that,
if resolved, would allow
a global vision of chaos.
One of them is to claim
that the situation we just encountered
should be typical.
In all dimensions
a typical vector field
should have the property that it has only
a finite number of attractors.
A typical initial condition
should be drawn to one of these attractors.
Each attractor should have
a Sinai-Ruelle-Bowen measure
that describes the asymptotic statistics
of the typical trajectories that it attracts.
A whole group of mathematicians
is hard at work on these conjectures,
and they are being solved
methodically, bit by bit.
A global picture seems to be emerging.
Is this picture is too optimistic?
Time will tell.
Today we no longer think of determinism
as the evolution of
an individual trajectory,
but rather as the
collective evolution of a whole set.
Sensitivity of trajectories to initial conditions
is compensated by a kind of
statistical stability of the whole set.
Did Buffon, in 1783
not foresee this
when he wrote this magnificent sentence
about a complex and chaotic world
that on has to try to understand globally?
"Everything happens,
because with time everything meets,
and in the free range of spaces
and in the continuous succession of movement,
all matter is stirred,
any form given,
all figures printed;
so everything is coming or going,
all is joined or running away,
all is combined or opposed,
everything happens or is destroyed
by relative or contrary forces,
that are the only constants,
and balancing without harm,
they animate the universe
and make it into a theater
with ever new scenes
and objects incessantly reborn ".