So, I wanted to introduce all of you
in the class
to Professor James Meiss who is in the Department
of applied mathematics at the University of Colorado
and my good colleague and downstairs neighbor.
Jim, I know you are particularly interested in
the standard map.
And we've been talking lately
in the...that I'm teaching about maps.
So can you tell us a little bit
about what the standard map is
and what is interesting to you about it.
Sure, I'm happy to talk about my favorite
dynamical system--the standard map.
So, my background is in physics
and the standard map is a good
model for some very simple physical systems
and one of them is
imagine that you have a pendulum
and ordinarily be rocking back and forth.
If you don't do anything to the pendulum
and there is no friction then it just rocks back and forth
the same amplitude forever and ever.
But if someone over here has a hammer.
and occasionally in this situation
gives it a kick periodically
So, every time the ball, wherever it happens to be
at t=1, the person hits it
and at t=2 the person will hit it again
and wherever it happens to be
the amplitude of the kick is fixed
but the hammer is more effective for different angles
than other angles
The kick will be more effective if say
the pendulum bob is horizontal
then you get a big kick.
And basically ineffective if it's straight down.
And give a negative kick if it's over here.
The physics of this tells you
just from Newton's equation
that you can reduce this dynamics to a map
a very simple map.
Here, the angle is represented by this variable x.
And, the angular momentum is represented
by the variable y.
And, so you get a two dimensional map that I've written
down totally wrong here.
There it is.
And, the dynamics of this system
is amazingly complicated
So if the kick is 0,
if parameter k is 0
then the angular momentum is constant.
And that's reflection of
the constant amplitude that I was telling you about.
But, if the k is nonzero than you have complicated
dynamics.
which sometimes is regular
and sometimes is chaotic
and I think the interested thing about this kind of system
which isn't dissipated is that there isn't just one.
typical attractor for this system
But is either strange or not strange
But, an inter mixture of some orbits
that's being chaotic and some orbits being regular
all through the same fixed parameter.
So, I have written a little program
that if you have a macintosh computer
you can download from my homepage.
If you click on the programs link here
There are several programs that I've written about
dynamical systems
One of them is the standard map program.
And, just download it by clicking on the link here.
It runs on most of the recent operating systems
for mac osx.
So, when you open the standard map program
you see two windows.
One is the phase space window
and the other is a text window
In the phase space based window
the horizontal variable is x
which is I showed you earlier is
the angle of the pendulum
and the vertical variable is y
the angular momentum.
And there is one parameter in the system
we can access by this map parameters menu called k
And, here I have just set parameter k to zero
as the initial trial.
If you click anywhere in the window
you see immediately horizontal lines in this situation.
These horizontal lines are actually consist
of many, many points.
So, if I instead of iterating continuously
go to single step and go to initial condition
you see a square where the initial condition is
Then hitting the space bar, shows you
one iterative of the map.
So, the analytical formula says that the
new value of x is the old value of x
plus whatever y we happen to be at.
Here the y is in the order of .3
So we move over to .3 and hit the space bar again
The colors change but the new point is again
the same distance horizontally over.
Since, this is an angle,
we wrap x around periodically.
Now we are over here some place.
Now we are over here some place.
Opps, I think I clicked some place incorrectly.
And if we keep iterating with the space bar.
It begins filling in the horizontal line
and if I go off to the find menu
and do continuously it just fills in a horizontal line
that indicates that initial condition
So changing the value of this parameter
k to something that is not zero
Something that is positive where we have a kick
We are kicking the system.
Now, if I click somewhere in the window.
The dynamical property of the orbit is quite different
So when you click, Jim, you are setting
an initial condition.
-Yes, Exactly. Setting an initial condition.
So, at different initial conditions have different behaviors.
So here the initial conditions is near x = 0,
which is where the pendulum would be oscillating
So, the range of x is bounded
and the momentum is positive and negative
So these are oscillatory orbits.
So up here, if I click with initial condition
the pendulum is going over the top.
And the interesting thing,
is there are orbits that have a hard time of deciding
if they are going to over the top
or oscillate.
And that's exactly where chaos starts to appear
in this dynamical system
So these are orbits on what we think of seperatrix
in the case of an integrable pendulum
this is an orbit where the pendulum starts with the bob
at the top. It falls down
crosses the down position
and goes back up vertically to the top
and the energy is preserved and it just asymptotic
to the top position and stays there
but when we are kicking it
the kick isn't very large here
but we are kicking it
and that results in orbit that seems to fit phase space
it seems to be chaotic
If I hit the up arrow key
I can just increment the value of K.
Here, the size of the chaotic region seems to grow.
Now, I mentioned before
is what is really interested about these conservative
dynamical systems
is I think there is an intermixture
of chaotic orbits and regular orbits
So the regular orbits that you see here
are these islands
And there are lots of islands in the system
and they are caused by
what we think of as resonances
in the Hamitonian dynamics
So there is a resonance here for example
that corresponds to three kicks per period
of the oscilation
-- And this kind is the resonance that causes planets
to have interesting behavior as well.
Yeah, so if we had a similar resonance
between the period of Jupiter and the period of an asteroid
Then, the asteroid, is effectively getting kicked
every time it goes near Jupiter and gets
kicked out of its orbit.
Another interesting thing about the standard map
is that there is a progression
of bifurcations that cause this chaotic region
to grow larger and larger
And if I make the parameter k,
and in this case let me type it in
on the order of say 5.
Then, the system becomes almost completely
chaotic
except for some small and irregular regions.
If we continue to increase the parameter value k
Those regions seem to be going away.
--Do they ever completely go away?
--No one knows.
It's been proved for any particular value of k
that's pretty large there are arbitrary close values
of k for which there are elliptic orbits.
Which are these islands.
But, it seems like if I set k on the order 10 say
the system as far as the computer is concerned
is completely chaotic.
We are filling one orbit here
and every pixel on the screen is gone.
It seems like the orbit fills the entire area.
But if i increase k to be 12 in this situation.
Suddenly an island--a visible island appears.
And there are visible islands that appear
for arbitrary large values of k.
--So a whole chain of bifurcations
but nothing like the period doubling
in the logistic map. This is much more complicated
--Yes, this is a different sequence of bifurcations.
--Another thing you might be interested in
on Jim's home page, is a link to the wonderful book
that he has written about dynamical systems.
--That's nice of you to say.
So, I wrote this book
for a class I teach for first year graduate students
in applied math and anybody in engineering or science
is interested in dynamics.
--Thanks Jim
--You are more welcomed.
In his lecture, Jim used a couple of terms
that we will talk about a little bit later into the semester
So, I just wanted to give you a forward pointer
and a quite explaination about those terms
First of all, Phase Space this is a synonym for state space.
State space is the very powerful representation
that we use to describe the evolution of trajectories
of dynamical systems.
It's the space of axis of the state variables.
For, the standard map, the state space
looks like this
And each point on that state space
The ones that jim was putting down by clicking on his app
represents a pair of values for x and y.
and the standard map which was invoking with the
click is a box of mathematics
that takes a value for x and a value for y
and tells you where the next point will be.
So, the action of the standard map is to evolve
the map one step at a time
as scanitized by that red arrow.
Of course the point doesn't actually follow
the path of the red arrow. It just hops from one dot
to the other.
A separatrix is what it sounds like
its a curve or surface in phase space
that separates two separate regions.
Kind of like the continental divide.
Integrable is a loaded word.
It basically means that the system is not chaotic
that the equations cannot be solved in closed form.
We will get back to this in the next unit.
This is another nest of words
that we will talk about later on
dissipation is a more general term
for friction.
Dissipation of energy, lost of energy
or technically a gain of energy.
In systems that are conservative
energy is conserved.
Those are non dissipative systems
Hamiltonian is a synonym for conservative
So, if I put a non in front of dissipative
those three words are synonyms.
Dissipation is a necessary condition for contractors.
You have to have dissipation
for an attractor in your system
because something has to eat up the energy
and cause the energy to relax down to the fixed point
where the chaotic cycle or limited attractor
So in the logistical map
where the trajectory is relaxing down to a fixed point
that behavior lets you know the system is dissipative
because there is an attractor there.
The standard map that Jim was talking about
is non dissipative, so there aren't any attractors.
But, there is still chaos
So just to sum that up.
Dissipation, friction, is a neccesary condition
for attractors
but not for the existence of chaos.
So, you can have chaos
if you don't have chaotic attractors
I'm not going to do a whole lot
of hamiltonian dynamics in this course
because we don't have time.
But it is a rich field
as Jim, I hope, suggested to you.