CHAOS 4- The swing
Simple things first...
In the sixteenth century
Galileo observed a swinging lamp,
and measured the oscillation period
by taking his own pulse.
How can we describe a swinging pendulum?
The tip of the pendulum
moves on the blue circle
Let’s place this circle on the cylinder,
like this
Every point of the circle
represents a position of the pendulum.
The red point for instance
is the position at the bottom.
Its speed,
the red vector,
is simply described by a number,
positive if it rotates in one direction
and negative in the other.
We can represent this number
in the cylinder’s vertical position.
So a single point on the cylinder
describes both the position and the speed.
The first coordinate, on the base circle,
tells us where the swing is
and the second, vertical,
tells us how fast it’s swinging.
So now,
let’s put gravity to work
and watch the swing!
For now
we assume that there is no friction at all!
The swinging will never end.
If the initial position is low
and if the initial velocity is not too large,
the pendulum moves periodically.
We all know that.
If we launch it faster
perhaps too fast,
it goes all the way round!
It is still a periodic movement
but not in the same way.
Look at the trajectory on the cylinder.
It follows a vector field...
Now observe the more realistic situation
where we assume that there is friction.
The oscillation slowly decreases,
and then finally
the pendulum tends to stop.
Remember Aristotle's theory:
the swing is now back
in its natural position at the bottom!
Observe the corresponding vector field
on the cylinder...
Trajectories eventually fall down,
to the resting position.
This equilibrium position
is called a stable attractor.
A swing with friction
is not so much fun
because it eventually stops
So we have to push the swing
to make it go really high.
This is also the point of view
of the gentleman
represented by the painter Fragonard
who seems interested in... the swing!
Let’s imagine that the pendulum,
in addition to gravity and friction,
is also subjected to a pushing force,
transmitted by a small nozzle.
A modern pendulum
with a small jet engine!
For example let’s push,
but only if the speed
and the inclination angle
are lower than a certain value.
If the swing starts off slowly,
it is pushed,
it accelerates, accelerates,
until the pushing stops
as it goes too fast.
Friction takes over and
it slows down, slows down,
until the push comes to the rescue again. etc..
The swing finally stabilizes
on a periodic regime.
Poincaré talks about a limit cycle.
The opposite of chaos, in a way.
The periodic regime.
Everything in sync!
Here is what is called
the phase portrait of the cylinder.
The limit cycle is in red.
We see a spiraling trajectory
that gradually becomes
the periodic trajectory.
Here is simple example
that illustrates all of this in ecology:
the Lotka-Volterra model,
dating from the 1930s.
Two populations share a territory.
These are for instance rabbits and foxes...
We will use our fantasy and
take ducks and water lilies.
Imagine that the ducks eat the water lilies.
When there aren’t many ducks
then they eat only few lilies.
The water lily population
therefore grows rapidly.
When there are few ducks
and many water lilies,
the ducks are well fed
and their number increases.
But a lot of ducks
eat a lot of water lilies!
When there are fewer and fewer water lilies
the ducks have nothing to eat
and their number decreases.
We have come full circle
and can start again!
We can show the situation at
a given point in time by a point on a plane:
the first coordinate represents
the number of water lilies
and the second the number of ducks.
In fact we get
a vector field in the plane...
Over time,
the ducks and lillies
follow a trajectory of this field and
reach a limit cycle.
The populations of ducks and water lilies
eventually oscillate periodically.
The belief that any motion,
perhaps after a short transition period,
eventually stabilizes,
either by stopping,
or by oscillating periodically,
has dominated science for a very long time.
One of the first theorems
in the theory of dynamical systems,
by Poincaré
in the late nineteenth century,
seemed to justify this.
It is about vector fields in the plane.
Imagine a vector field like this:
In fact, we don’t know the vector field everywhere,
but we know how it behaves
close to this circle.
Take a trajectory that enters the disk.
Where can it go?
Well,
The Poincaré-Bendixson theorem
says that there are two possible cases:
The trajectory
will either get very close to
an equilibrium position:
as we see here...
this is what we saw
for the damped pendulum
- or it must approach a limit cycle.
How can we prove such a theorem?
Here is the main idea.
Take a point on the circle and
observe its trajectory
that enters the circle..
Stop!
Let’s stop at a point P!
Look at the trajectories
in the vicinity of P.
Is it possible that the trajectory
continues and returns the vicinity of P?
Let’s suppose that it indeed
returns to a point Q.
The arc of the trajectory between P and Q,
followed by the line QP
forms a closed curve.
This is the boundary of
a certain area, drawn in blue.
We can see that
a trajectory that starts from Q
enters the blue area
and then can’t get out anymore!
It’s trapped...
In order to exit,
it should cross the arc PQ
but two trajectories cannot cross.
Why?
Well if two trajectories
passed through the same point,
this would contradict the
Cauchy – Lipschitz theorem: there is only
a single trajectory through any given point!
The trajectory starting from Q
cannot escape through the line QP either
as there the vector field is
coming in, not going out.
You can see that the trajectory from P
may well come back
not too far from P
but then it is condemned
to never return...
It is said that there is no recurrence:
This is the main idea of the
Poincaré-Bendixson theorem.
This theorem marks the beginning
of what is called
the qualitative theory of dynamical systems.
Even if one has only an imperfect
knowledge of a vector field,
one can often understand
the behavior of its trajectories.
In the case of the plane,
everything goes well:
the trajectories become
periodic in time or they
approach a position of equilibrium.
But Poincaré soon discovered
that his theorem is valid only
for fields in two dimensions,
that is to say for very small systems...
From three dimensions on up,
we will see that the situation can be much,
much, much richer... and pretty!
No more nice limit cycles,
welcome to the world of chaos!